PARI reference
Every PARI function, its ER2 name, and whether the prelude exposes it. Generated from er2/data/pari_functions.csv.
PARI functions and their ER2 names
Generated by tools/sync_pari_functions.py from the libpari bundled with cypari2 (PARI 2.17.2). Do not edit by hand: edit er2/data/pari_functions.csv and re-run the script.
| Status | Meaning | Count |
|---|---|---|
prelude |
top-level ER2 name (in the prelude) | 31 |
namespace |
available as pari.<er2_name> |
946 |
wrapper |
needs an ER2 wrapper (takes GP code; not a cypari2 method) | 26 |
python |
not exposed: Python or its ecosystem already covers it | 165 |
conflict |
not exposed yet: naming conflict to resolve | 0 |
| total | 1168 |
1. Programming in GP (126)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
addhelp |
python | use Python | add/change help message for the symbol sym | |
alarm |
python | use Python | if code is omitted, trigger an “e_ALARM” exception after s seconds (wall-clock time), cancelling any previously set alar… | |
alias |
python | use Python | defines the symbol newsym as an alias for the symbol sym | |
allocatemem |
python | use Python | allocates a new stack of s bytes | |
apply |
python | use Python | apply function f to each entry in A | |
arity |
python | use Python | return the arity of the closure C | |
break |
python | use Python | interrupt execution of current instruction sequence, and exit from the n innermost enclosing loops | |
breakpoint |
python | use Python | interrupt the program and enter the breakloop | |
call |
python | use Python | A being a vector, evaluates f(A[1],…,A[#A]) | |
dbg_x |
python | use Python | print inner structure of A, complete if n is omitted, up to level n otherwise | |
default |
python | use Python | returns the current value of the default key | |
errname |
python | use Python | returns the type of the error message E | |
error |
python | use Python | abort script with error message str | |
export |
python | use Python | export the variables x,…,z to the parallel world | |
exportall |
python | use Python | declare all current dynamic variables as exported variables | |
extern |
python | use Python | execute shell command str, and feeds the result to GP (as if loading from file) | |
externstr |
python | use Python | execute shell command str, and returns the result as a vector of GP strings, one component per output line | |
fileclose |
python | use Python | close the file descriptor n | |
fileextern |
python | use Python | execute shell command str and returns a file descriptor attached to the command output as if it were read from a file | |
fileflush |
python | use Python | flush the file descriptor n (all descriptors to output streams if n is omitted) | |
fileopen |
python | use Python | open the file pointed to by ‘path’ and return a file descriptor which can be used with other file functions | |
fileread |
python | use Python | read a logical line from the file attached to the descriptor n, opened for reading with fileopen | |
filereadstr |
python | use Python | read a raw line from the file attached to the descriptor n, opened for reading with fileopen | |
filewrite |
python | use Python | write the string s to file attached to descriptor n, ending with a newline | |
filewrite1 |
python | use Python | write the string s to file number n without ending with newline | |
fold |
python | use Python | return f(…f(f(A[1],A[2]),A[3]),…,A[#A]) | |
for |
python | use Python | the sequence is evaluated, X going from a up to b | |
forcomposite |
python | use Python | the sequence is evaluated, n running over the composite numbers between a and b | |
fordiv |
python | use Python | the sequence is evaluated, X running over the divisors of n | |
fordivfactored |
python | use Python | the sequence is evaluated, X running over the [d, factor(d)], d a divisor of n | |
foreach |
python | use Python | the sequence is evaluated, X running over the components of V | |
forell |
python | use Python | execute seq for each elliptic curves E of conductor between a and b in the elldata database | |
forfactored |
python | use Python | the sequence is evaluated, N is of the form [n, factor(n)], n going from a up to b | |
forpart |
python | use Python | evaluate seq where the Vecsmall X goes over the partitions of k | |
forperm |
python | use Python | the sequence is evaluated, p going through permutations of a | |
forprime |
python | use Python | the sequence is evaluated, p running over the primes between a and b | |
forprimestep |
python | use Python | the sequence is evaluated, p running over the primes less than b in the arithmetic progression a + k*q, k >= 0 | |
forsquarefree |
python | use Python | the sequence is evaluated, N is of the form [n, factor(n)], n going through squarefree integers from a up to b | |
forstep |
python | use Python | the sequence is evaluated, X going from a to b in steps of s (can be a positive real number, an intmod for an arithmetic… | |
forsubgroup |
python | use Python | execute seq for each subgroup H of the abelian group G, whose index is bounded by bound if not omitted | |
forsubset |
python | use Python | if nk is an integer n, the sequence is evaluated, s going through all subsets of {1, 2, …, n}; if nk is a pair [n,k] o… | |
forvec |
python | use Python | v being a two-component vectors of length n, the sequence is evaluated with X[i] going from v[i][1] to v[i][2] for i=n,.… | |
getabstime |
python | use Python | milliseconds of CPU time since startup | |
getcache |
python | use Python | returns information about various auto-growing caches | |
getenv |
python | use Python | value of the environment variable s, 0 if it is not defined | |
getheap |
python | use Python | 2-component vector giving the current number of objects in the heap and the space they occupy (in long words) | |
getlocalbitprec |
python | use Python | returns the current dynamic bit precision | |
getlocalprec |
python | use Python | returns the current dynamic precision, in decimal digits | |
getrand |
python | use Python | current value of random number seed | |
getstack |
python | use Python | current value of stack pointer avma | |
gettime |
python | use Python | milliseconds of CPU time used since the last call to gettime | |
getwalltime |
python | use Python | time (in milliseconds) since the UNIX Epoch | |
global |
python | use Python | obsolete | |
if |
python | use Python | if a is nonzero, seq1 is evaluated, otherwise seq2 | |
iferr |
python | use Python | evaluates the expression sequence seq1 | |
inline |
python | use Python | declares x,…,z as inline variables | |
input |
python | use Python | read an expression from the input file or standard input | |
install |
python | use Python | load from dynamic library ‘lib’ the function ‘name’ | |
kill |
python | use Python | restores the symbol sym to its `undefined'' status and kill attached help messages | |listcreate| | python | use Python | this function is obsolete, use List() | |listinsert| | python | use Python | insert x at index n in list L, shifting the remaining elements to the right | |listkill| | python | use Python | obsolete, retained for backward compatibility | |listpop| | python | use Python | removes n-th element from list | |listput| | python | use Python | sets n-th element of list equal to x | |listsort| | python | use Python | sort the list L in place | |local| | python | use Python | declare x,...,z as (dynamically scoped) local variables | |localbitprec| | python | use Python | set the real precision to p bits in the dynamic scope | |localprec| | python | use Python | set the real precision to p in the dynamic scope and return p | |mapapply| | python | use Python | applies the closure f to the image of x by the map M and returns the evaluation of f | |mapdelete| | python | use Python | removes x from the domain of the map M | |mapget| | python | use Python | returns the image of x by the map M | |mapisdefined| | python | use Python | true (1) if x has an image by the map M, false (0) otherwise | |mapput| | python | use Python | associates x to y in the map M | |my| | python | use Python | declare x,...,z as lexically-scoped local variables | |next| | python | use Python | interrupt execution of current instruction sequence, and start another iteration from the n-th innermost enclosing loops | |parapply| | python | use Python | parallel evaluation of f on the elements of x | |pareval| | python | use Python | parallel evaluation of the elements of the vector of closures x | |parfor| | python | use Python | evaluates the expression expr1 in parallel for all i between a and b (if b is set to +oo, the loop will not stop), resul… | |parforeach| | python | use Python | evaluates in parallel the expression expr1 for all components x of V | |parforprime| | python | use Python | evaluates the expression expr1 in parallel for all primes p between a and b (if b is set to +oo, the loop will not stop)… | |parforprimestep| | python | use Python | evaluates the expression expr1 in parallel for all primes p between a and b in an arithmetic progression of the form a +… | |parforstep| | python | use Python | evaluates the expression expr1 in parallel for i going from a to b in steps of s (can be a positive real number, an intm… | |parforvec| | python | use Python | evaluates the sequence expr2 (dependent on X and j) for X as generated by forvec, in random order, computed in parallel | |parselect| | python | use Python | (parallel select) selects elements of A according to the selection function f which is tested in parallel | |parsum| | python | use Python | the sum (i goes from a to b) of expression expr, evaluated in parallel (in random order) | |parvector| | python | use Python | as vector(N,i,expr) but the evaluations of expr are done in parallel | |print| | python | use Python | outputs its string arguments (in raw format) ending with a newline | |print1| | python | use Python | outputs its string arguments (in raw format) without ending with newline | |printf| | python | use Python | prints its arguments according to the format fmt | |printp| | python | use Python | outputs its string arguments (in prettymatrix format) ending with a newline | |printsep| | python | use Python | outputs its string arguments (in raw format), separated by 'sep', ending with a newline | |printsep1| | python | use Python | outputs its string arguments (in raw format), separated by 'sep', without ending with a newline | |printtex| | python | use Python | outputs its string arguments in TeX format | |read| | python | use Python | read from the input file filename | |readstr| | python | use Python | returns the vector of GP strings containing the lines in filename | |readvec| | python | use Python | create a vector whose components are the evaluation of all the expressions found in the input file filename | |return| | python | use Python | return from current subroutine with result x | |select| | python | use Python | selects elements of A according to the selection function f | |self| | python | use Python | return the calling function or closure | |setdebug| | python | use Python | sets debug level for domain D to n (n must be between 0 and 20) | |setrand| | python | use Python | reset the seed of the random number generator to n | |Strchr| | python | use Python | deprecated alias for strchr | |strchr| | python | use Python | converts integer or vector of integers x to a string, translating each integer into a character using ASCII encoding | |Strexpand| | python | use Python | deprecated alias for strexpand | |strexpand| | python | use Python | concatenates its (string) arguments into a single string, performing tilde expansion | |strjoin| | python | use Python | joins the strings in vector v, separating them with delimiter p | |Strprintf| | python | use Python | deprecated alias for strprintf | |strprintf| | python | use Python | returns a string built from the remaining arguments according to the format fmt | |strsplit| | python | use Python | splits the string s into a vector of strings, with p acting as a delimiter between successive fields; if p is empty or o… | |Strtex| | python | use ER2 latex() (§3.6) | deprecated alias for strtex | |strtex| | python | use ER2 latex() (§3.6) | translates its (string) arguments to TeX format and returns the resulting string | |strtime| | python | use Python | return a string describing the time t in milliseconds, in the format used by the GP timer | |system| | python | use Python | str being a string, execute the system command str | |trap| | python | use Python | this function is obsolete, use "iferr" | |type| | python | use Python | return the type of the GEN x | |unexport| | python | use Python | remove x,...,z from the list of variables exported to the parallel world | |unexportall| | python | use Python | empty the list of variables exported to the parallel world | |uninline| | python | use Python | forget all inline variables | |until| | python | use Python | evaluate the expression sequence seq until a is nonzero | |version| | python | use Python | returns the PARI version as [major,minor,patch] or [major,minor,patch,GITversion] | |warning| | python | use Python | display warning message str | |while| | python | use Python | while a is nonzero evaluate the expression sequence seq | |write| | python | use Python | appends the remaining arguments (same output as print) to filename | |write1| | python | use Python | appends the remaining arguments (same output as print1) to filename | |writebin| | python | use Python | write x as a binary object to file filename | |writetex` |
2. Standard operators (10)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
cmp |
cmp |
namespace | compare two arbitrary objects x and y (1 if x>y, 0 if x=y, -1 if x<y) | |
divrem |
divrem |
namespace | euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainder, with respect to v (… | |
lex |
lex |
namespace | compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x<y) | |
max |
max |
namespace | Python builtin: never shadowed in the prelude | maximum of x and y |
min |
min |
namespace | Python builtin: never shadowed in the prelude | minimum of x and y |
shift |
shift |
namespace | shift x left n bits if n>=0, right -n bits if n<0 | |
shiftmul |
shiftmul |
namespace | multiply x by 2^n (n>=0 or n<0) | |
sign |
sign |
namespace | sign of x, of type integer, real or fraction | |
vecmax |
vecmax |
namespace | largest entry in the vector/matrix x | |
vecmin |
vecmin |
namespace | smallest entry in the vector/matrix x |
3. Conversions and elementary functions (60)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
binary |
binary |
namespace | gives the vector formed by the binary digits of x (x integer) | |
bitand |
bitand |
namespace | bitwise “and” of two integers x and y | |
bitneg |
bitneg |
namespace | bitwise negation of an integers x truncated to n bits | |
bitnegimply |
bitnegimply |
namespace | bitwise “negated imply” of two integers x and y, in other words, x BITAND BITNEG(y) | |
bitor |
bitor |
namespace | bitwise “or” of two integers x and y | |
bitprecision |
bitprecision |
namespace | if n is present and positive, return x at precision n bits | |
bittest |
bittest |
namespace | gives bit number n (coefficient of 2^n) of the integer x | |
bitxor |
bitxor |
namespace | bitwise “exclusive or” of two integers x and y | |
ceil |
ceil |
namespace | ceiling of x = smallest integer >= x | |
centerlift |
centerlift |
namespace | centered lift of x | |
characteristic |
characteristic |
namespace | characteristic of the base ring over which x is defined | |
Col |
Col |
namespace | type constructor (CapWords allowed by PEP 8) | transforms the object x into a column vector of dimension n |
Colrev |
Colrev |
namespace | type constructor (CapWords allowed by PEP 8) | transforms the object x into a column vector of dimension n in reverse order with respect to Col(x,{n}) |
component |
component |
namespace | the n’th component of the internal representation of x | |
conj |
conj |
namespace | the algebraic conjugate of x | |
conjvec |
conjvec |
namespace | conjugate vector of the algebraic number z | |
denominator |
denominator |
namespace | denominator of f | |
digits |
digits |
namespace | gives the vector formed by the digits of x in base b | |
exponent |
exponent |
namespace | binary exponent of x | |
floor |
floor |
namespace | floor of x = largest integer <= x | |
frac |
frac |
namespace | fractional part of x = x-floor(x) | |
fromdigits |
fromdigits |
namespace | gives the integer formed by the elements of x seen as the digits of a number in base b | |
imag |
imag |
namespace | imaginary part of x | |
length |
length |
namespace | number of non code words in x, number of characters for a string | |
lift |
lift |
namespace | if v is omitted, lifts elements of Z/nZ to Z, of Qp to Q, and of K[x]/(P) to K[x] | |
liftall |
liftall |
namespace | lifts every element of Z/nZ to Z, of Qp to Q, and of K[x]/(P) to K[x] | |
liftint |
liftint |
namespace | lifts every element of Z/nZ to Z and of Qp to Q | |
liftpol |
liftpol |
namespace | lifts every polmod component of x to polynomials | |
List |
List |
namespace | type constructor (CapWords allowed by PEP 8) | transforms the vector or list x into a list |
Map |
Map |
namespace | type constructor (CapWords allowed by PEP 8) | converts the matrix [a_1,b_1;a_2,b_2;…;a_n,b_n] to the map a_i->b_i |
Mat |
Mat |
namespace | type constructor (CapWords allowed by PEP 8) | transforms any GEN x into a matrix |
Mod |
Mod |
namespace | type constructor (CapWords allowed by PEP 8) | create ‘a modulo b’ |
norm |
norm |
namespace | norm of x | |
numerator |
numerator |
namespace | numerator of f | |
oo |
oo |
namespace | infinity | |
padicprec |
padicprec |
namespace | return the absolute p-adic precision of object x | |
Pol |
Pol |
namespace | type constructor (CapWords allowed by PEP 8) | convert t (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficient |
Polrev |
Polrev |
namespace | type constructor (CapWords allowed by PEP 8) | convert t (usually a vector or a power series) into a polynomial with variable v, starting with the constant term |
precision |
precision |
namespace | if n is present, return x at precision n | |
Qfb |
Qfb |
namespace | type constructor (CapWords allowed by PEP 8) | binary quadratic form a*x2+bxy+c*y2 |
random |
random |
namespace | random object, depending on the type of N | |
real |
real |
namespace | real part of x | |
round |
round |
namespace | Python builtin: never shadowed in the prelude | take the nearest integer to all the coefficients of x |
Ser |
Ser |
namespace | type constructor (CapWords allowed by PEP 8) | convert s into a power series with variable v and precision d, starting with the constant coefficient |
serchop |
serchop |
namespace | remove all terms of degree strictly less than n in series s | |
serprec |
serprec |
namespace | return the absolute precision x with respect to power series in the variable v | |
Set |
Set |
namespace | type constructor (CapWords allowed by PEP 8) | convert x into a set, i.e |
simplify |
simplify |
namespace | simplify the object x as much as possible | |
sizebyte |
sizebyte |
namespace | number of bytes occupied by the complete tree of the object x | |
sizedigit |
sizedigit |
namespace | rough upper bound for the number of decimal digits of (the components of) x | |
Str |
python | Python’s str covers it | concatenates its (string) argument into a single string | |
truncate |
truncate |
namespace | truncation of x; when x is a power series,take away the O(X^) | |
valuation |
valuation |
prelude | valuation of x with respect to p | |
varhigher |
varhigher |
namespace | return a variable ‘name’ whose priority is higher than the priority of v (of all existing variables if v is omitted) | |
variable |
variable |
namespace | main variable of object x | |
variables |
variables |
namespace | all variables occurring in object x, sorted by decreasing priority | |
varlower |
varlower |
namespace | return a variable ‘name’ whose priority is lower than the priority of v (of all existing variables if v is omitted | |
Vec |
Vec |
namespace | type constructor (CapWords allowed by PEP 8) | transforms the object x into a vector of dimension n |
Vecrev |
Vecrev |
namespace | type constructor (CapWords allowed by PEP 8) | transforms the object x into a vector of dimension n in reverse order with respect to Vec(x,{n}) |
Vecsmall |
Vecsmall |
namespace | type constructor (CapWords allowed by PEP 8) | transforms the object x into a VECSMALL of dimension n |
4. Combinatorics (21)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
bernfrac |
bernfrac |
namespace | Bernoulli number B_n, as a rational number | |
bernpol |
bernpol |
namespace | Bernoulli polynomial B_n, evaluated at a | |
bernreal |
bernreal |
namespace | Bernoulli number B_n, as a real number with the current precision | |
bernvec |
bernvec |
namespace | returns a vector containing, as rational numbers, the Bernoulli numbers B_0, B_2, …, B_{2n} | |
binomial |
binomial |
prelude | binomial coefficient n(n-1)…(n-k+1)/k! defined for k in Z and any n | |
eulerfrac |
eulerfrac |
namespace | Euler number E_n, as a rational number | |
eulerianpol |
eulerianpol |
namespace | Eulerian polynomial A_n, in variable v | |
eulerpol |
eulerpol |
namespace | Euler polynomial E_n, in variable v | |
eulerreal |
eulerreal |
namespace | Euler number E_n, as a real number | |
eulervec |
eulervec |
namespace | returns a vector containing the nonzero Euler numbers E_0, E_2, …, E_{2n} | |
fibonacci |
fibonacci |
prelude | Fibonacci number of index x | |
hammingweight |
hammingweight |
namespace | returns the Hamming weight of x | |
harmonic |
harmonic |
namespace | generalized harmonic number of index n in power r | |
numbpart |
numbpart |
namespace | number of partitions of n | |
numtoperm |
numtoperm |
namespace | permutation number k (mod n!) of n letters (n C-integer) | |
partitions |
partitions |
namespace | vector of partitions of the integer k | |
permcycles |
permcycles |
namespace | cycles of the permutation x | |
permorder |
permorder |
namespace | order of the permutation x | |
permsign |
permsign |
namespace | signature of the permutation x | |
permtonum |
permtonum |
namespace | ordinal (between 0 and n!-1) of permutation x | |
stirling |
stirling |
namespace | if flag=1 (default) return the Stirling number of the first kind s(n,k), if flag=2, return the Stirling number of the se… |
5. Number theory (127)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
addprimes |
addprimes |
namespace | add primes in the vector x to the prime table to be used in trial division | |
bestappr |
bestappr |
namespace | return a rational approximation to x, whose denominator is limited by B, if present | |
bestapprPade |
bestappr_pade |
namespace | renamed for PEP 8 | returns a rational function approximation to x |
bezout |
bezout |
namespace | deprecated alias for gcdext | |
bigomega |
bigomega |
prelude | draft Omega, renamed for PEP 8 (D8) |
number of prime divisors of x, counted with multiplicity |
charconj |
charconj |
namespace | given a finite abelian group (by its elementary divisors cyc) and a character chi, return the conjugate character | |
chardiv |
chardiv |
namespace | given a finite abelian group (by its elementary divisors cyc) and two characters a and b, return the character a/b | |
chareval |
chareval |
namespace | given an abelian group structure affording a discrete logarithm method, e.g | |
chargalois |
chargalois |
namespace | let cyc represent a finite abelian group G by its elementary divisors cyc, return a list of representatives for the Galo… | |
charker |
charker |
namespace | given a finite abelian group (by its elementary divisors cyc) and a character chi, return its kernel | |
charmul |
charmul |
namespace | given a finite abelian group (by its elementary divisors cyc) and two characters a and b, return the product character a… | |
charorder |
charorder |
namespace | given a finite abelian group (by its elementary divisors cyc) and a character chi, return the order of chi | |
charpow |
charpow |
namespace | given a finite abelian group (by its elementary divisors cyc) a character a and an integer n return the character a^n | |
chinese |
chinese |
prelude | x,y being both intmods (or polmods) computes z in the same residue classes as x and y | |
content |
content |
namespace | gcd of all the components of x, when this makes sense | |
contfrac |
contfrac |
namespace | continued fraction expansion of x (x rational,real or rational function) | |
contfracpnqn |
contfracpnqn |
namespace | [p_n,p_{n-1}; q_n,q_{n-1}] corresponding to the continued fraction x | |
core |
core |
prelude | unique squarefree integer d dividing n such that n/d is a square | |
coredisc |
coredisc |
namespace | discriminant of the quadratic field Q(sqrt(n)) | |
dirdiv |
dirdiv |
namespace | division of the Dirichlet series x by the Dirichlet series y | |
direuler |
direuler |
wrapper | GP expression argument; ER2 takes a Python callable | Dirichlet Euler product of expression expr from p=a to p=b, limited to b terms |
dirmul |
dirmul |
namespace | multiplication of the Dirichlet series x by the Dirichlet series y | |
dirpowerssum |
dirpowerssum |
namespace | return f(1)1^x + f(2)2^x + | |
divisors |
divisors |
prelude | gives a vector formed by the divisors of x in increasing order | |
divisorslenstra |
divisorslenstra |
namespace | finds all divisors d of N such that d = r (mod s) | |
eulerphi |
phi |
prelude | draft name | Euler’s totient function of x |
factor |
factor |
prelude | dispatch: integers -> PARI, Expr -> SymPy | factorization of x over domain D |
factorback |
factorback |
namespace | given a factorization f, gives the factored object back | |
factorcantor |
factorcantor |
namespace | this function is obsolete, use factormod | |
factorff |
factorff |
namespace | obsolete, use factormod | |
factorial |
factorial |
namespace | PARI returns a real; the exact factorial is in the prelude |
factorial of x, the result being given as a real number |
factorint |
factorint |
namespace | factor the integer x | |
factormod |
factormod |
namespace | ER2: factor(f, modulus=p) |
factors the polynomial f over the finite field defined by the domain D; flag is optional, and can be 0: default or 1: on… |
factormodcyclo |
factormodcyclo |
namespace | factor n-th cyclotomic polynomial mod p | |
factormodDDF |
factormod_ddf |
namespace | renamed for PEP 8 | distinct-degree factorization of the squarefree polynomial f over the finite field defined by the domain D |
factormodSQF |
factormod_sqf |
namespace | renamed for PEP 8 | squarefree factorization of the polynomial f over the finite field defined by the domain D |
ffcompomap |
ffcompomap |
namespace | Let k, l, m be three finite fields and f a (partial) map from l to m and g a partial map from k to l, return the (partia… | |
ffembed |
ffembed |
namespace | given two elements a and b in finite fields, return a map embedding the definition field of a to the definition field of… | |
ffextend |
ffextend |
namespace | extend the field K of definition of a by a root of the polynomial P, assumed to be irreducible over K | |
fffrobenius |
fffrobenius |
namespace | return the n-th power of the Frobenius map over the field of definition of m | |
ffgen |
ffgen |
namespace | ER2: GF(q).gen() |
return a generator of the finite field k (not necessarily a generator of its multiplicative group) as a t_FFELT |
ffinit |
ffinit |
namespace | ER2: GF(p, k) |
monic irreducible polynomial of degree n in F_p[v] |
ffinvmap |
ffinvmap |
namespace | given a map m between finite fields, return a partial map that return the pre-images by the map m | |
fflog |
fflog |
namespace | return the discrete logarithm of the finite field element x in base g | |
ffmap |
ffmap |
namespace | given a (partial) map m between two finite fields, return the image of x by m | |
ffmaprel |
ffmaprel |
namespace | given a (partial) map m between two finite fields, express x as an algebraic element over the codomain of m in a way whi… | |
ffnbirred |
ffnbirred |
namespace | number of monic irreducible polynomials over F_q, of degree n (flag=0, default) or at most n (flag=1) | |
fforder |
fforder |
namespace | multiplicative order of the finite field element x | |
ffprimroot |
ffprimroot |
namespace | return a primitive root of the multiplicative group of the definition field of the finite field element x (not necessari… | |
gcd |
gcd |
prelude | greatest common divisor of x and y | |
gcdext |
gcdext |
namespace | returns [u,v,d] such that d=gcd(x,y) and ux+vy=d | |
halfgcd |
halfgcd |
namespace | return a vector [M, [a,b]~], where M is an invertible 2x2 matrix such that M*[x,y]~ = [a,b]~, where b is small | |
hilbert |
hilbert |
namespace | Hilbert symbol at p of x,y | |
isfundamental |
isfundamental |
namespace | true(1) if D is a fundamental discriminant (including 1), false(0) if not | |
ispolygonal |
ispolygonal |
namespace | true(1) if x is an s-gonal number, false(0) if not (s > 2) | |
ispower |
ispower |
prelude | if k > 0 is given, return true (1) if x is a k-th power, false (0) if not | |
ispowerful |
ispowerful |
namespace | true(1) if x is a powerful integer (valuation at all primes dividing x is greater than 1), false(0) if not | |
isprime |
isprime |
prelude | true(1) if x is a (proven) prime number, false(0) if not | |
isprimepower |
isprimepower |
prelude | if x = p^k is a prime power (p prime, k > 0), return k, else return 0 | |
ispseudoprime |
ispseudoprime |
prelude | true(1) if x is a strong pseudoprime, false(0) if not | |
ispseudoprimepower |
ispseudoprimepower |
namespace | if x = p^k is a pseudo-prime power (p pseudo-prime, k > 0), return k, else return 0 | |
issquare |
issquare |
prelude | true(1) if x is a square, false(0) if not | |
issquarefree |
issquarefree |
prelude | true(1) if x is squarefree, false(0) if not | |
istotient |
istotient |
namespace | true(1) if x = eulerphi(n) for some integer n, false(0) if not | |
kronecker |
kronecker |
prelude | kronecker symbol (x/y) | |
lcm |
lcm |
prelude | least common multiple of x and y, i.e | |
logint |
logint |
namespace | return the largest non-negative integer e so that b^e <= x, where b > 1 is an integer and x >= 1 is a real number | |
moebius |
mu |
prelude | draft name | Moebius function of x |
nextprime |
nextprime |
prelude | smallest pseudoprime >= x | |
numdiv |
numdiv |
prelude | number of divisors of x | |
omega |
omega |
prelude | number of distinct prime divisors of x | |
precprime |
precprime |
prelude | largest pseudoprime <= x, 0 if x<=1 | |
prime |
prime |
prelude | returns the n-th prime (n C-integer) | |
primecert |
primecert |
namespace | If N is a prime, return a Primality Certificate | |
primecertexport |
primecertexport |
namespace | Returns a string suitable for print/write to display a primality certificate | |
primecertisvalid |
primecertisvalid |
namespace | Verifies if cert is a valid PARI ECPP Primality certificate | |
primepi |
primepi |
prelude | the prime counting function pi(x) = #{p <= x, p prime} | |
primes |
primes |
prelude | returns the vector of the first n primes (integer), or the primes in interval n = [a,b] | |
qfbclassno |
qfbclassno |
namespace | class number of discriminant D using Shanks’s method by default | |
qfbcomp |
qfbcomp |
namespace | Gaussian composition with reduction of the binary quadratic forms x and y | |
qfbcompraw |
qfbcompraw |
namespace | Gaussian composition without reduction of the binary quadratic forms x and y | |
qfbcornacchia |
qfbcornacchia |
namespace | Solves the equation x2+dy2 = n in integers x and y where d > 0 and n is prime or 4 times a prime | |
qfbhclassno |
qfbhclassno |
namespace | Hurwitz-Kronecker class number of x>0 | |
qfbnucomp |
qfbnucomp |
namespace | composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where L=[|D/4|^(1/4)] is preco… | |
qfbnupow |
qfbnupow |
namespace | n-th power of primitive positive definite quadratic form x using nucomp and nudupl | |
qfbpow |
qfbpow |
namespace | n-th power with reduction of the binary quadratic form x | |
qfbpowraw |
qfbpowraw |
namespace | n-th power without reduction of the binary quadratic form x | |
qfbprimeform |
qfbprimeform |
namespace | returns the prime form of discriminant x, whose first coefficient is p | |
qfbred |
qfbred |
namespace | reduction of the binary quadratic form x | |
qfbredsl2 |
qfbredsl2 |
namespace | reduction of the binary quadratic form x, returns [y,g] where y is reduced and g in Sl(2,Z) is such that g.x = y; isD, i… | |
qfbsolve |
qfbsolve |
namespace | Solve the equation Q(x,y)=n in coprime integers x and y where Q is a binary quadratic form, up to the action of the spec… | |
quadclassunit |
quadclassunit |
namespace | compute the structure of the class group and the regulator of the quadratic field of discriminant D | |
quaddisc |
quaddisc |
namespace | discriminant of the quadratic field Q(sqrt(x)) | |
quadgen |
quadgen |
namespace | standard generator g of quadratic order of discriminant D | |
quadhilbert |
quadhilbert |
namespace | relative equation for the Hilbert class field of the quadratic field of discriminant D (which can also be a bnf) | |
quadpoly |
quadpoly |
namespace | quadratic polynomial corresponding to the discriminant D, in variable v | |
quadray |
quadray |
namespace | relative equation for the ray class field of conductor f for the quadratic field of discriminant D (which can also be a … | |
quadregulator |
quadregulator |
namespace | regulator of the real quadratic field of discriminant D | |
quadunit |
quadunit |
namespace | fundamental unit u of the quadratic order of discriminant D where D must be positive | |
quadunitindex |
quadunitindex |
namespace | given a fundamental discriminant D, returns the index of the unit group of the order of conductor f | |
quadunitnorm |
quadunitnorm |
namespace | returns the norm of the fundamental unit of the quadratic order of discriminant D | |
ramanujantau |
ramanujantau |
namespace | compute the value of Ramanujan’s tau function at n, assuming the GRH | |
randomprime |
randomprime |
namespace | returns a strong pseudo prime in [2, N-1] | |
removeprimes |
removeprimes |
namespace | remove primes in the vector x from the prime table | |
sigma |
sigma |
prelude | sum of the k-th powers of the divisors of x | |
sqrtint |
sqrtint |
prelude | integer square root y of x, where x is a nonnegative real number | |
sqrtnint |
sqrtnint |
namespace | integer n-th root of x, where x is nonnegative real number | |
sumdedekind |
sumdedekind |
namespace | Dedekind sum attached to h,k | |
sumdigits |
sumdigits |
namespace | sum of digits in the integer n, when written in base B | |
znchar |
znchar |
namespace | given a datum D describing a group G = (Z/NZ)^* and a Dirichlet character chi, return the pair [G,chi] | |
zncharconductor |
zncharconductor |
namespace | let G be znstar(q,1) and chi be a Dirichlet character on (Z/qZ)* | |
znchardecompose |
znchardecompose |
namespace | given a znstar G = (Z/NZ)^* and a Dirichlet character chi, return the product of local characters chi_p for p | (N,Q) | |
znchargauss |
znchargauss |
namespace | given a Dirichlet character chi on G = (Z/NZ)^*, return the complex Gauss sum g(chi,a) | |
zncharinduce |
zncharinduce |
namespace | let G be znstar(q,1), let chi be a Dirichlet character mod q and let N be a multiple of q | |
zncharisodd |
zncharisodd |
namespace | let G be znstar(N,1), let chi be a Dirichlet character mod N, return 1 if and only if chi(-1) = -1 and 0 otherwise | |
znchartokronecker |
znchartokronecker |
namespace | let G be znstar(N,1), let chi be a Dirichlet character mod N, return the discriminant D if chi is real equal to the Kron… | |
znchartoprimitive |
znchartoprimitive |
namespace | let G be znstar(q,1) and chi be a Dirichlet character on (Z/qZ)* of conductor q0 | |
znconreychar |
znconreychar |
namespace | Dirichlet character attached to m in (Z/qZ)* in Conrey’s notation, where G is znstar(q,1) | |
znconreyconductor |
znconreyconductor |
namespace | let G be znstar(q,1) and chi be a Dirichlet character on (Z/qZ)* given by its Conrey logarithm | |
znconreyexp |
znconreyexp |
namespace | Conrey exponential attached to G = znstar(q, 1) | |
znconreylog |
znconreylog |
namespace | Conrey logarithm attached to m in (Z/qZ)*, where G is znstar(q,1) | |
zncoppersmith |
zncoppersmith |
namespace | finds all integers x with |x| <= X such that gcd(N, P(x)) >= B | |
znlog |
znlog |
prelude | return the discrete logarithm of x in (Z/nZ)* in base g | |
znorder |
znorder |
prelude | order of the integermod x in (Z/nZ)* | |
znprimroot |
znprimroot |
prelude | returns a primitive root of n when it exists | |
znstar |
znstar |
namespace | 3-component vector v = [no,cyc,gen], giving the structure of the abelian group (Z/nZ)^*; no is the order (i.e | |
znsubgroupgenerators |
znsubgroupgenerators |
namespace | finds generators of the subgroup H of (Z/fZ)^*; H is given by a vector of length f of 1/0 values: the a-th component is … |
6. Polynomials and power series (65)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
bezoutres |
bezoutres |
namespace | deprecated alias for polresultantext | |
deriv |
deriv |
namespace | ER2 diff dispatches to SymPy |
derivative of x with respect to v, or to the main variable of x if v is omitted |
derivn |
derivn |
namespace | n-th derivative of x with respect to v, or to the main variable of x if v is omitted | |
diffop |
diffop |
namespace | apply the differential operator D to x, where D is defined by D(v[i])=d[i], where v is a vector of variable names | |
eval |
python | GP code evaluation; pari.raw evaluates GP code if needed | evaluation of x, replacing variables by their value | |
factorpadic |
factorpadic |
namespace | p-adic factorization of the polynomial pol to precision r | |
fft |
fft |
namespace | given w from rootsof1, return the discrete Fourier transform of P | |
fftinv |
fftinv |
namespace | given w from rootsof1, return the inverse Fourier transform of P | |
intformal |
intformal |
namespace | formal integration of x with respect to v, or to the main variable of x if v is omitted | |
O |
python | use SymPy’s O or series(); PARI series convert both ways | p-adic or power series zero with precision given by e | |
padicappr |
padicappr |
namespace | p-adic roots of the polynomial pol congruent to a mod p | |
padicfields |
padicfields |
namespace | returns polynomials generating all the extensions of degree N of the field of p-adic rational numbers; N is allowed to b… | |
polchebyshev |
polchebyshev |
namespace | Chebyshev polynomial of the first (flag = 1) or second (flag = 2) kind, of degree n, evaluated at a | |
polclass |
polclass |
namespace | return a polynomial generating the Hilbert class field of Q(sqrt(D)) for the discriminant D<0 | |
polcoef |
polcoef |
namespace | coefficient of degree n of x | |
polcoeff |
polcoeff |
namespace | deprecated alias for polcoef | |
polcyclo |
polcyclo |
namespace | n-th cyclotomic polynomial evaluated at a | |
polcyclofactors |
polcyclofactors |
namespace | returns a vector of polynomials, whose product is the product of distinct cyclotomic polynomials dividing f | |
poldegree |
poldegree |
namespace | degree of the polynomial or rational function x with respect to main variable if v is omitted, with respect to v otherwi… | |
poldisc |
poldisc |
namespace | ER2: discriminant |
discriminant of the polynomial pol, with respect to main variable if v is omitted, with respect to v otherwise |
poldiscfactors |
poldiscfactors |
namespace | [D, faD], where D = discriminant of the polynomial T, and faD is a cheap partial factorization of D (entries are coprime… | |
poldiscreduced |
poldiscreduced |
namespace | vector of elementary divisors of Z[a]/f’(a)Z[a], where a is a root of the polynomial f | |
polfromroots |
polfromroots |
namespace | returns the monic polynomial in variable v whose roots are the components of the vector a with multiplicities | |
polgraeffe |
polgraeffe |
namespace | returns the Graeffe transform g of f, such that g(x^2) = f(x)f(-x) | |
polhensellift |
polhensellift |
namespace | lift the factorization B of A modulo p to a factorization modulo p^e using Hensel lift | |
polhermite |
polhermite |
namespace | Hermite polynomial H(n,v) of degree n, evaluated at a | |
polinterpolate |
polinterpolate |
namespace | polynomial interpolation at t according to data vectors X, Y, i.e., given P of minimal degree such that P(X[i]) = Y[i] f… | |
polisclass |
polisclass |
namespace | P being a monic irreducible polynomial with integer coefficients, return 0 if P is not a class polynomial for the j-inva… | |
poliscyclo |
poliscyclo |
namespace | returns 0 if f is not a cyclotomic polynomial, and n > 0 if f = Phi_n, the n-th cyclotomic polynomial | |
poliscycloprod |
poliscycloprod |
namespace | returns 1 if f is a product of cyclotomic polynonials, and 0 otherwise | |
polisirreducible |
polisirreducible |
namespace | ER2: isirreducible |
true(1) if pol is an irreducible nonconstant polynomial, false(0) if pol is reducible or constant |
pollaguerre |
pollaguerre |
namespace | Laguerre polynomial of degree n and parameter a evaluated at b | |
pollead |
pollead |
namespace | leading coefficient of polynomial or series x, or x itself if x is a scalar | |
pollegendre |
pollegendre |
namespace | legendre polynomial of degree n evaluated at a | |
polmodular |
polmodular |
namespace | return the modular polynomial of level L and invariant inv | |
polrecip |
polrecip |
namespace | reciprocal polynomial of pol | |
polresultant |
polresultant |
namespace | ER2: resultant |
resultant of the polynomials x and y, with respect to the main variables of x and y if v is omitted, with respect to the… |
polresultantext |
polresultantext |
namespace | return [U,V,R] such that R=polresultant(A,B,v) and UA+VB = R, where A and B are polynomials | |
polroots |
polroots |
namespace | complex roots of the polynomial T using Schonhage’s method, as modified by Gourdon | |
polrootsbound |
polrootsbound |
namespace | return a sharp upper bound for the modulus of the largest complex root of the polynomial T with relative error tau | |
polrootsff |
polrootsff |
namespace | obsolete, use polrootsmod | |
polrootsmod |
polrootsmod |
namespace | roots of the polynomial f over the finite field defined by the domain D | |
polrootspadic |
polrootspadic |
namespace | p-adic roots of the polynomial f to precision r | |
polrootsreal |
polrootsreal |
namespace | real roots of the polynomial T with real coefficients, using Uspensky’s method | |
polsturm |
polsturm |
namespace | number of distinct real roots of the polynomial T (in the interval ab = [a,b] if present) | |
polsubcyclo |
polsubcyclo |
namespace | finds an equation (in variable v) for the d-th degree subfields of Q(zeta_n) | |
polsubcyclofast |
polsubcyclofast |
namespace | If 1 <= d <= 6 or a prime, finds an equation for the subfields of Q(zeta_n) with galois group C_d | |
polsylvestermatrix |
polsylvestermatrix |
namespace | forms the sylvester matrix attached to the two polynomials x and y | |
polsym |
polsym |
namespace | column vector of symmetric powers of the roots of x up to n | |
poltchebi |
poltchebi |
namespace | deprecated alias for polchebyshev | |
polteichmuller |
polteichmuller |
namespace | return the polynomial whose roots (resp | |
poltomonic |
poltomonic |
namespace | T in Q[x]; returns U monic in Z[x] such that U(x) = C T(x/L) for some rational C and L | |
polzagier |
polzagier |
namespace | Zagier’s polynomials of index n,m | |
seralgdep |
seralgdep |
namespace | find a linear relation between powers (1,s, …, s^p) of the series s, with polynomial coefficients of degree <= r | |
serconvol |
serconvol |
namespace | convolution (or Hadamard product) of two power series | |
serdiffdep |
serdiffdep |
namespace | find an inhomogenous linear differential equation satisfied by the series s, with polynomial coefficients of degree <= r | |
serlaplace |
serlaplace |
namespace | replaces the power series sum of a_nx^n/n! by sum of a_nx^n | |
serreverse |
serreverse |
namespace | reversion of the power series s | |
subst |
subst |
namespace | in expression x, replace the variable y by the expression z | |
substpol |
substpol |
namespace | in expression x, replace the polynomial y by the expression z, using remainder decomposition of x | |
substvec |
substvec |
namespace | in expression x, make a best effort to replace the variables v1,…,vn by the expression w1,…,wn | |
sumformal |
sumformal |
namespace | formal sum of f with respect to v, or to the main variable of f if v is omitted | |
taylor |
taylor |
namespace | taylor expansion of x with respect to t, adding O(t^d) to all components of x | |
thue |
thue |
namespace | solve the equation P(x,y)=a, where tnf was created with thueinit(P), and sol, if present, contains the solutions of Norm… | |
thueinit |
thueinit |
namespace | initialize the tnf corresponding to P, that will be used to solve Thue equations P(x,y) = some-integer |
7. Linear algebra (91)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
algdep |
algdep |
namespace | algebraic relations up to degree k of z, using lindep([1,z,…,z^(k-1)], flag) | |
bestapprnf |
bestapprnf |
namespace | T being an integral polynomial and V being a scalar, vector, or matrix, return a reasonable approximation of V with polm… | |
charpoly |
charpoly |
namespace | det(v*Id-A)=characteristic polynomial of the matrix or polmod A | |
concat |
concat |
namespace | concatenation of x and y, which can be scalars, vectors or matrices, or lists (in this last case, both x and y have to b… | |
dirpowers |
dirpowers |
namespace | return the vector [1x,2x,…,n^x] | |
forqfvec |
forqfvec |
wrapper | GP expression argument; ER2 takes a Python callable | q being a square and symmetric integral matrix representing an positive definite quadratic form, evaluate expr for all p… |
lindep |
lindep |
namespace | integral linear dependencies between components of v | |
matadjoint |
matadjoint |
namespace | adjoint matrix of M using Leverrier-Faddeev’s algorithm | |
matcompanion |
matcompanion |
namespace | companion matrix to polynomial x | |
matconcat |
matconcat |
namespace | concatenate the entries of v and return the resulting matrix | |
matdet |
matdet |
namespace | ER2: det |
determinant of the matrix x using an appropriate algorithm depending on the coefficients |
matdetint |
matdetint |
namespace | some multiple of the determinant of the lattice generated by the columns of B (0 if not of maximal rank) | |
matdetmod |
matdetmod |
namespace | determinant of the matrix x modulo d | |
matdiagonal |
matdiagonal |
namespace | creates the diagonal matrix whose diagonal entries are the entries of the vector x | |
mateigen |
mateigen |
namespace | complex eigenvectors of the matrix x given as columns of a matrix H | |
matfrobenius |
matfrobenius |
namespace | return the Frobenius form of the square matrix M | |
mathess |
mathess |
namespace | Hessenberg form of x | |
mathilbert |
mathilbert |
namespace | Hilbert matrix of order n | |
mathnf |
mathnf |
namespace | ER2: hermite_form |
(upper triangular) Hermite normal form of M, basis for the lattice formed by the columns of M |
mathnfmod |
mathnfmod |
namespace | (upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, where d is a multiple of … | |
mathnfmodid |
mathnfmodid |
namespace | (upper triangular) Hermite normal form of x concatenated with matdiagonal(d) | |
mathouseholder |
mathouseholder |
namespace | applies a sequence Q of Householder transforms to the vector or matrix v | |
matid |
matid |
namespace | identity matrix of order n | |
matimage |
matimage |
namespace | basis of the image of the matrix x | |
matimagecompl |
matimagecompl |
namespace | vector of column indices not corresponding to the indices given by the function matimage | |
matimagemod |
matimagemod |
namespace | basis of the image of the matrix x modulo d | |
matindexrank |
matindexrank |
namespace | gives two extraction vectors (rows and columns) for the matrix M such that the extracted matrix is square of maximal ran… | |
matintersect |
matintersect |
namespace | intersection of the vector spaces whose bases are the columns of x and y | |
matinverseimage |
matinverseimage |
namespace | an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwise | |
matinvmod |
matinvmod |
namespace | left inverse of the matrix x modulo d | |
matisdiagonal |
matisdiagonal |
namespace | true(1) if x is a diagonal matrix, false(0) otherwise | |
matker |
matker |
namespace | ER2: kernel |
basis of the kernel of the matrix x |
matkerint |
matkerint |
namespace | LLL-reduced Z-basis of the kernel of the matrix x with integral entries; flag is deprecated, kept for backward compatibi… | |
matkermod |
matkermod |
namespace | basis of the kernel of the matrix x modulo d | |
matmuldiagonal |
matmuldiagonal |
namespace | product of matrix x by diagonal matrix whose diagonal coefficients are those of the vector d, equivalent but faster than… | |
matmultodiagonal |
matmultodiagonal |
namespace | product of matrices x and y, knowing that the result will be a diagonal matrix | |
matpascal |
matpascal |
namespace | Pascal triangle of order n if q is omitted | |
matpermanent |
matpermanent |
namespace | permanent of the matrix x | |
matqr |
matqr |
namespace | returns [Q,R], the QR-decomposition of the square invertible matrix M | |
matrank |
matrank |
namespace | ER2: rank |
rank of the matrix x |
matreduce |
matreduce |
namespace | reduce the factorization matrix m to canonical form (sorted first row with unique elements) matrix | |
matrix |
matrix |
namespace | m x n matrix of expression expr, where the row variable X goes from 1 to m and the column variable Y goes from 1 to n | |
matrixqz |
matrixqz |
namespace | if p>=0, transforms the rational or integral mxn (m>=n) matrix A into an integral matrix with gcd of maximal determinant… | |
matsize |
matsize |
namespace | number of rows and columns of the vector/matrix x as a 2-vector | |
matsnf |
matsnf |
namespace | ER2: smith_form |
Smith normal form (i.e |
matsolve |
matsolve |
namespace | ER2: solve(A, b) |
solution of MX=B (M matrix, B column vector or matrix) |
matsolvemod |
matsolvemod |
namespace | one solution of system of congruences MX=B mod D (M matrix, B and D column vectors) | |
matsupplement |
matsupplement |
namespace | supplement the columns of the matrix x to an invertible matrix | |
mattranspose |
mattranspose |
namespace | x~ = transpose of x | |
minpoly |
minpoly |
namespace | minimal polynomial of the matrix or polmod A | |
norml2 |
norml2 |
namespace | square of the L2-norm of x | |
normlp |
normlp |
namespace | Lp-norm of x; sup norm if p is omitted | |
powers |
powers |
namespace | return the vector [1,x,…,x^n] if x0 is omitted, and [x0, x0x, …, x0x^n] otherwise | |
qfauto |
qfauto |
namespace | automorphism group of the positive definite quadratic form G | |
qfautoexport |
qfautoexport |
namespace | qfa being an automorphism group as output by qfauto, output a string representing the underlying matrix group in GAP not… | |
qfbil |
qfbil |
namespace | this function is obsolete, use qfeval | |
qfcholesky |
qfcholesky |
namespace | given a square symmetric matrix M, return R such that R~*R = M, or [] if there is no solution | |
qfcvp |
qfcvp |
namespace | x being a square and symmetric matrix representing a positive definite quadratic form, and t a vector of the same dimens… | |
qfeval |
qfeval |
namespace | evaluate the quadratic form q (symmetric matrix) at x; if y is present, evaluate the polar form at (x,y); if q omitted, … | |
qfgaussred |
qfgaussred |
namespace | square reduction of the symmetric matrix q | |
qfisom |
qfisom |
namespace | find an isomorphism between the integral positive definite quadratic forms G and H if it exists | |
qfisominit |
qfisominit |
namespace | G being a square and symmetric matrix representing an integral positive definite quadratic form, this function returns a… | |
qfjacobi |
qfjacobi |
namespace | eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix A | |
qflll |
qflll |
namespace | LLL reduction of the vectors forming the matrix x (gives the unimodular transformation matrix T such that x*T is LLL-red… | |
qflllgram |
qflllgram |
namespace | LLL reduction of the lattice whose gram matrix is G (gives the unimodular transformation matrix) | |
qfminim |
qfminim |
namespace | x being a square and symmetric matrix representing a positive definite quadratic form, this function deals with the vect… | |
qfminimize |
qfminimize |
namespace | given a square symmetric matrix G with rational coefficients and non-zero determinant, of dimension n >= 1, return [H,U,… | |
qfnorm |
qfnorm |
namespace | this function is obsolete, use qfeval | |
qforbits |
qforbits |
namespace | return the orbits of V under the action of the group of linear transformation generated by the set G, which must stabili… | |
qfparam |
qfparam |
namespace | coefficients of binary quadratic forms that parametrize the solutions of the ternary quadratic form G, using the particu… | |
qfperfection |
qfperfection |
namespace | rank of matrix of xx~ for x minimal vectors of a Gram matrix G | |
qfrep |
qfrep |
namespace | vector of (half) the number of vectors of norms from 1 to B for the integral and definite quadratic form q | |
qfsign |
qfsign |
namespace | signature of the symmetric matrix x | |
qfsolve |
qfsolve |
namespace | solve over Q the quadratic equation X~ G X = 0, where G is a symmetric matrix | |
setbinop |
setbinop |
namespace | the set {f(x,y), x in X, y in Y} | |
setdelta |
setdelta |
namespace | symmetric difference of the sets x and y | |
setintersect |
setintersect |
namespace | intersection of the sets x and y | |
setisset |
setisset |
namespace | true(1) if x is a set (row vector with strictly increasing entries), false(0) if not | |
setminus |
setminus |
namespace | set of elements of x not belonging to y | |
setsearch |
setsearch |
namespace | determines whether x belongs to the set (or sorted list) S | |
setunion |
setunion |
namespace | union of the sets x and y | |
snfrank |
snfrank |
namespace | assuming that D is a Smith normal form (i.e | |
trace |
trace |
namespace | trace of x | |
vecextract |
vecextract |
namespace | extraction of the components of the matrix or vector x according to y and z | |
vecprod |
vecprod |
namespace | return the product of the components of the vector v | |
vecsearch |
vecsearch |
namespace | determines whether x belongs to the sorted vector v | |
vecsort |
vecsort |
namespace | sorts the vector x in ascending order, according to the comparison function cmpf, if not omitted | |
vecsum |
vecsum |
namespace | return the sum of the components of the vector v | |
vector |
vector |
namespace | row vector with n components of expression expr (X ranges from 1 to n) | |
vectorsmall |
vectorsmall |
wrapper | GP expression argument; ER2 takes a Python callable | VECSMALL with n components of expression expr (X ranges from 1 to n) which must be small integers |
vectorv |
vectorv |
wrapper | GP expression argument; ER2 takes a Python callable | column vector with n components of expression expr (X ranges from 1 to n) |
8. Transcendental functions (73)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
abs |
abs |
namespace | Python builtin: never shadowed in the prelude | absolute value (or modulus) of x |
acos |
acos |
namespace | arc cosine of x | |
acosh |
acosh |
namespace | inverse hyperbolic cosine of x | |
agm |
agm |
namespace | arithmetic-geometric mean of x and y | |
airy |
airy |
namespace | Airy [Ai,Bi] function of argument z | |
arg |
arg |
namespace | argument of x, such that -pi<arg(x)<=pi | |
asin |
asin |
namespace | arc sine of x | |
asinh |
asinh |
namespace | inverse hyperbolic sine of x | |
atan |
atan |
namespace | arc tangent of x | |
atanh |
atanh |
namespace | inverse hyperbolic tangent of x | |
besselh1 |
besselh1 |
namespace | H^1-bessel function of index nu and argument x | |
besselh2 |
besselh2 |
namespace | H^2-bessel function of index nu and argument x | |
besseli |
besseli |
namespace | I-bessel function of index nu and argument x | |
besselj |
besselj |
namespace | J-bessel function of index nu and argument x | |
besseljh |
besseljh |
namespace | J-bessel function of index n+1/2 and argument x, where n is a nonnegative integer | |
besseljzero |
besseljzero |
namespace | k-th zero of the J-bessel function of index nu | |
besselk |
besselk |
namespace | K-bessel function of index nu and argument x | |
besseln |
besseln |
namespace | deprecated alias for bessely | |
bessely |
bessely |
namespace | Y-bessel function of index nu and argument x | |
besselyzero |
besselyzero |
namespace | k-th zero of the Y-bessel function of index nu | |
Catalan |
catalan |
namespace | constant; SymPy name | Catalan’s number with current precision |
cos |
cos |
namespace | cosine of x | |
cosh |
cosh |
namespace | hyperbolic cosine of x | |
cotan |
cotan |
namespace | cotangent of x | |
cotanh |
cotanh |
namespace | hyperbolic cotangent of x | |
dilog |
dilog |
namespace | dilogarithm of x | |
eint1 |
eint1 |
namespace | exponential integral E1(x) | |
ellE |
ell_e |
namespace | renamed for PEP 8 | Complete elliptic integral of the second kind for the complex parameter k using the agm |
ellK |
ell_k |
namespace | renamed for PEP 8 | Complete elliptic integral of the first kind for the complex parameter k using the agm |
erfc |
erfc |
namespace | complementary error function | |
eta |
eta |
namespace | if flag=0, returns prod(n=1,oo, 1-q^n), where q = exp(2 i Pi z) if z is a complex scalar (belonging to the upper half pl… | |
Euler |
euler_gamma |
namespace | constant; SymPy name | Euler’s constant with current precision |
exp |
exp |
namespace | exponential of x | |
expm1 |
expm1 |
namespace | exp(x)-1 | |
gamma |
gamma |
namespace | gamma function at s, a complex or p-adic number, or a series | |
gammah |
gammah |
namespace | gamma of x+1/2 (x integer) | |
gammamellininv |
gammamellininv |
namespace | returns G(t), where G is as output by gammamellininvinit (its m-th derivative if m is present) | |
gammamellininvasymp |
gammamellininvasymp |
namespace | return the first n terms of the asymptotic expansion at infinity of the m-th derivative K^m(t) of the inverse Mellin tra… | |
gammamellininvinit |
gammamellininvinit |
namespace | initialize data for the computation by gammamellininv() of the m-th derivative of the inverse Mellin transform of the fu… | |
hypergeom |
hypergeom |
namespace | general hypergeometric function, where N and D are the vector of parameters in the numerator and denominator respectivel… | |
hyperu |
hyperu |
namespace | U-confluent hypergeometric function | |
I |
I |
namespace | constant; SymPy name | square root of -1 |
incgam |
incgam |
namespace | incomplete gamma function | |
incgamc |
incgamc |
namespace | complementary incomplete gamma function | |
lambertw |
lambertw |
namespace | solution of the implicit equation x*exp(x)=y | |
lerchphi |
lerchphi |
namespace | Lerch transcendent equal to sum for n >= 0 of z^n / (n+a)^s for reasonable values of the arguments | |
lerchzeta |
lerchzeta |
namespace | Lerch zeta function equal to sum for n >= 0 of e^(2 pi i lam n) / (n+a)^s for reasonable values of the arguments | |
lngamma |
lngamma |
namespace | logarithm of the gamma function of x | |
log |
log |
namespace | natural logarithm of x | |
log1p |
log1p |
namespace | log(1+x) | |
Pi |
pi |
namespace | constant; lowercase like SymPy | the constant pi, with current precision |
polylog |
polylog |
namespace | m-th polylogarithm of x | |
polylogmult |
polylogmult |
namespace | multiple polylogarithm value at integral s = [s1,…,sr] with argument z = [z1,…,zr] | |
psi |
digamma |
namespace | PARI psi = digamma; Dedekind psi is dedekind_psi (D5) | psi-function at x (der-th derivative of psi if der is set) |
rootsof1 |
rootsof1 |
namespace | column vector of complex N-th roots of 1 | |
sin |
sin |
namespace | sine of x | |
sinc |
sinc |
namespace | sinc function of x | |
sinh |
sinh |
namespace | hyperbolic sine of x | |
sqr |
sqr |
namespace | square of x | |
sqrt |
sqrt |
namespace | square root of x | |
sqrtn |
sqrtn |
namespace | nth-root of x, n must be integer | |
tan |
tan |
namespace | tangent of x | |
tanh |
tanh |
namespace | hyperbolic tangent of x | |
teichmuller |
teichmuller |
namespace | Teichmuller character of p-adic number x | |
theta |
theta |
namespace | Jacobi sine theta-function | |
thetanullk |
thetanullk |
namespace | k-th derivative at z=0 of theta(q,z) | |
weber |
weber |
namespace | one of Weber’s f function of x | |
zeta |
zeta |
namespace | Riemann zeta function at s with s a complex or a p-adic number | |
zetahurwitz |
zetahurwitz |
namespace | Hurwitz zeta function at s, x, with s not 1 and x not a negative or zero integer | |
zetamult |
zetamult |
namespace | multiple zeta value at integral s = [s1,…,sk]; more generally, return Yamamoto’s t-MZV interpolation (star value for t… | |
zetamultall |
zetamultall |
namespace | list of all multiple zeta values for weight up to k | |
zetamultconvert |
zetamultconvert |
namespace | a being either an evec, avec, or index m, converts into evec (flag=0), avec (flag=1), or index m (flag=2) | |
zetamultdual |
zetamultdual |
namespace | s being either an evec, avec, or index m, return the dual sequence in avec format |
9. Sums, products, integrals (39)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
asympnum |
asympnum |
namespace | asymptotic expansion of expr assuming it has rational coefficients with reasonable height; alpha is as in limitnum | |
asympnumraw |
asympnumraw |
namespace | N+1 first terms of asymptotic expansion of expr as floating point numbers; alpha is as in limitnum | |
contfraceval |
contfraceval |
namespace | given a continued fraction CF from contfracinit, evaluate the first lim terms of the continued fraction at t (all terms … | |
contfracinit |
contfracinit |
namespace | given M representing the power series S = sum_{n>=0} M[n+1]z^n, transform it into a continued fraction suitable for eval… | |
derivnum |
derivnum |
wrapper | GP expression argument; ER2 takes a Python callable | numerical derivation of expr with respect to X at X = a |
intcirc |
intcirc |
wrapper | GP expression argument; ER2 takes a Python callable | numerical integration of expr on the circle |z-a|=R, divided by 2IPi |
intfuncinit |
python | builds internal integration tables; not exposed | initialize tables for integrations from a to b using a weight f(t) | |
intnum |
intnum |
wrapper | GP expression argument; ER2 takes a Python callable | numerical integration of expr from a to b with respect to X |
intnumgauss |
intnumgauss |
wrapper | GP expression argument; ER2 takes a Python callable | numerical integration of expr from a to b, a compact interval, with respect to X using Gauss-Legendre quadrature |
intnumgaussinit |
intnumgaussinit |
namespace | initialize tables for n-point Gauss-Legendre integration on a compact interval | |
intnuminit |
intnuminit |
namespace | initialize tables for integrations from a to b | |
intnumosc |
intnumosc |
wrapper | GP expression argument; ER2 takes a Python callable | numerical integration from a to oo of oscillating quasi-periodic function expr of half-period H |
intnumromb |
intnumromb |
wrapper | GP expression argument; ER2 takes a Python callable | numerical integration of expr (smooth in ]a,b[) from a to b with respect to X |
laurentseries |
laurentseries |
namespace | expand f around 0 as a Laurent series in x to order M | |
limitnum |
limitnum |
namespace | numerical limit of sequence expr using Lagrange-Zagier extrapolation; assume u(n) ~ sum a_i n^(-alpha*i) | |
prod |
prod |
wrapper | GP expression argument; ER2 takes a Python callable | x times the product (X runs from a to b) of expression |
prodeuler |
prodeuler |
wrapper | GP expression argument; ER2 takes a Python callable | Euler product (p runs over the primes between a and b) of real or complex expression, as a floating point approximation |
prodeulerrat |
prodeulerrat |
namespace | product from primes p = a to infinity of F(p^s), where F is a rational function | |
prodinf |
prodinf |
wrapper | GP expression argument; ER2 takes a Python callable | infinite product (X goes from a to infinity) of real or complex expression |
prodnumrat |
prodnumrat |
namespace | product from n = a to infinity of F(n), where F-1 is a rational function of degree less than or equal to -2 | |
solve |
solve |
wrapper | GP expression argument; ER2 takes a Python callable | real root of expression expr (X between a and b), where either a or b is infinite or expr(a)*expr(b)<=0 |
solvestep |
solvestep |
wrapper | GP expression argument; ER2 takes a Python callable | find zeros of a function in the real interval [a,b] by naive interval splitting |
sum |
sum |
wrapper | Python builtin: never shadowed in the prelude; GP expression argument; ER2 takes a Python callable | x plus the sum (X goes from a to b) of expression expr |
sumalt |
sumalt |
wrapper | GP expression argument; ER2 takes a Python callable | Cohen-Villegas-Zagier’s acceleration of alternating series expr, X starting at a |
sumdiv |
sumdiv |
wrapper | GP expression argument; ER2 takes a Python callable | sum of expression expr, X running over the divisors of n |
sumdivmult |
sumdivmult |
wrapper | GP expression argument; ER2 takes a Python callable | sum of multiplicative function expr, d running over the divisors of n |
sumeulerrat |
sumeulerrat |
namespace | sum from primes p = a to infinity of F(p^s), where F is a rational function | |
suminf |
suminf |
wrapper | GP expression argument; ER2 takes a Python callable | naive summation (X goes from a to infinity) of real or complex expression expr |
sumnum |
sumnum |
wrapper | GP expression argument; ER2 takes a Python callable | numerical summation of f(n) from n = a to +infinity using Euler-MacLaurin summation |
sumnumap |
sumnumap |
wrapper | GP expression argument; ER2 takes a Python callable | numerical summation of f(n) from n = a to +infinity using Abel-Plana formula |
sumnumapinit |
sumnumapinit |
namespace | initialize tables for Abel-Plana summation of a series | |
sumnuminit |
sumnuminit |
namespace | initialize tables for Euler-MacLaurin delta summation of a series with positive terms | |
sumnumlagrange |
sumnumlagrange |
wrapper | GP expression argument; ER2 takes a Python callable | numerical summation of f(n) from n = a to +infinity using Lagrange summation |
sumnumlagrangeinit |
sumnumlagrangeinit |
namespace | initialize tables for Lagrange summation of a series | |
sumnummonien |
sumnummonien |
wrapper | GP expression argument; ER2 takes a Python callable | numerical summation from n = a to +infinity using Monien summation |
sumnummonieninit |
sumnummonieninit |
namespace | initialize tables for Monien summation of a series with positive terms | |
sumnumrat |
sumnumrat |
namespace | sum from n = a to infinity of F(n), where F is a rational function of degree less than or equal to -2 | |
sumnumsidi |
sumnumsidi |
wrapper | GP expression argument; ER2 takes a Python callable | numerical summation of f(n) from n = a to +infinity using Sidi summation; a must be an integer |
sumpos |
sumpos |
wrapper | GP expression argument; ER2 takes a Python callable | sum of positive (or negative) series expr, the formal variable X starting at a |
10. Number fields (210)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
bnfcertify |
bnfcertify |
namespace | ER2: certify=True |
certify the correctness (i.e |
bnfdecodemodule |
bnfdecodemodule |
namespace | given a coded module m as in bnrdisclist, gives the true module | |
bnfinit |
bnfinit |
namespace | ER2: NumberField class group and units |
compute the necessary data for future use in ideal and unit group computations, including fundamental units if they are … |
bnfisintnorm |
bnfisintnorm |
namespace | compute a complete system of solutions (modulo units of positive norm) of the absolute norm equation N(a)=x, where a bel… | |
bnfisnorm |
bnfisnorm |
namespace | tries to tell whether x (in Q) is the norm of some fractional y (in bnf) | |
bnfisprincipal |
bnfisprincipal |
namespace | bnf being output by bnfinit, gives [e,t], where e is the vector of exponents on the class group generators and t is the … | |
bnfissunit |
bnfissunit |
namespace | this function is obsolete, use bnfisunit | |
bnfisunit |
bnfisunit |
namespace | bnf being output by bnfinit, give the column vector of exponents of x on the fundamental units and the roots of unity if… | |
bnflog |
bnflog |
namespace | let bnf be attached to a number field F and let l be a prime number | |
bnflogdegree |
bnflogdegree |
namespace | let A be an ideal, return exp(deg_F A) the exponential of the l-adic logarithmic degree | |
bnflogef |
bnflogef |
namespace | return [e~, f~] the logarithmic ramification and residue degrees for the maximal ideal pr | |
bnfnarrow |
bnfnarrow |
namespace | given a big number field as output by bnfinit, gives as a 3-component vector the structure of the narrow class group | |
bnfsignunit |
bnfsignunit |
namespace | matrix of signs of the real embeddings of the system of fundamental units found by bnfinit | |
bnfsunit |
bnfsunit |
namespace | compute the fundamental S-units of the number field bnf output by bnfinit, S being a list of prime ideals | |
bnfunits |
bnfunits |
namespace | return the fundamental units of the number field bnf output by bnfinit; if S is present and is a list of prime ideals, c… | |
bnrchar |
bnrchar |
namespace | returns all characters chi on G such that chi(g[i]) = e(v[i]); if v is omitted, returns all characters that are trivial … | |
bnrclassfield |
bnrclassfield |
namespace | bnr being as output by bnrinit, find a relative equation for the class field corresponding to the congruence subgroup de… | |
bnrclassno |
bnrclassno |
namespace | relative degree of the class field defined by A,B,C | |
bnrclassnolist |
bnrclassnolist |
namespace | if list is as output by ideallist or similar, gives list of corresponding ray class numbers | |
bnrcompositum |
bnrcompositum |
namespace | compositum [bnr,H] of the two abelian extensions given by A = [bnr1,H1] and B = [bnr2,H2], where bnr1 and bnr2 are attac… | |
bnrconductor |
bnrconductor |
namespace | conductor f of the subfield of the ray class field given by A,B,C | |
bnrconductorofchar |
bnrconductorofchar |
namespace | this function is obsolete, use bnrconductor | |
bnrdisc |
bnrdisc |
namespace | absolute or relative [N,R1,discf] of the field defined by A,B,C | |
bnrdisclist |
bnrdisclist |
namespace | list of discriminants of ray class fields of all conductors up to norm bound | |
bnrgaloisapply |
bnrgaloisapply |
namespace | apply the automorphism given by its matrix mat to the congruence subgroup H given as a HNF matrix | |
bnrgaloismatrix |
bnrgaloismatrix |
namespace | return the matrix of the action of the automorphism aut of the base field bnf.nf on the generators of the ray class fiel… | |
bnrinit |
bnrinit |
namespace | given a bnf as output by bnfinit and a modulus f, initializes data linked to the ray class group structure corresponding… | |
bnrisconductor |
bnrisconductor |
namespace | returns 1 if the modulus is the conductor of the subfield of the ray class field given by A,B,C (see bnrdisc), and 0 oth… | |
bnrisgalois |
bnrisgalois |
namespace | check whether the class field attached to the subgroup H is Galois over the subfield of bnr.nf fixed by the Galois group… | |
bnrisprincipal |
bnrisprincipal |
namespace | bnr being output by bnrinit and x being an ideal coprime to bnr.mod, returns [v,alpha], where v is the vector of exponen… | |
bnrL1 |
bnr_l1 |
namespace | renamed for PEP 8 | bnr being output by bnrinit and H being a square matrix defining a congruence subgroup of bnr (the trivial subgroup if o… |
bnrmap |
bnrmap |
namespace | if A and B are bnr structures for the same bnf attached to moduli mA and mB with mB | mA, return the canonical surjectio… | |
bnrrootnumber |
bnrrootnumber |
namespace | returns the so-called Artin Root Number, i.e | |
bnrstark |
bnrstark |
namespace | bnr being as output by bnrinit, finds a relative equation for the class field corresponding to the module in bnr and the… | |
bnrstarkunit |
bnrstarkunit |
namespace | bnr being as output by bnrinit, returns the characteristic polynomial of the (conjectural) Stark unit corresponding to t… | |
dirzetak |
dirzetak |
namespace | Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1 | |
factornf |
factornf |
namespace | this function is obsolete, use nffactor | |
galoischardet |
galoischardet |
namespace | return the determinant character of the character chi | |
galoischarpoly |
galoischarpoly |
namespace | return the list of characteristic polynomials of the representation attached to the character chi | |
galoischartable |
galoischartable |
namespace | return the character table of the underlying group of gal | |
galoisconjclasses |
galoisconjclasses |
namespace | gal being output by galoisinit, return the list of conjugacy classes | |
galoisexport |
galoisexport |
namespace | gal being a Galois group as output by galoisinit, output a string representing the underlying permutation group in GAP n… | |
galoisfixedfield |
galoisfixedfield |
namespace | gal being a Galois group as output by galoisinit and perm a subgroup, an element of gal.group or a vector of such elemen… | |
galoisgetgroup |
galoisgetgroup |
namespace | query the galpol package for a group of order a with index b in the GAP4 Small Group library | |
galoisgetname |
galoisgetname |
namespace | query the galpol package for a string describing the group of order a with index b in the GAP4 Small Group library | |
galoisgetpol |
galoisgetpol |
namespace | query the galpol package for a polynomial with Galois group isomorphic to GAP4(a,b), totally real if s=1 (default) and t… | |
galoisidentify |
galoisidentify |
namespace | gal being a Galois group as output by galoisinit, output the isomorphism class of the underlying abstract group as a two… | |
galoisinit |
galoisinit |
namespace | pol being a polynomial or a number field as output by nfinit defining a Galois extension of Q, compute the Galois group … | |
galoisisabelian |
galoisisabelian |
namespace | gal being as output by galoisinit, return 0 if gal is not abelian, the HNF matrix of gal over gal.gen if flag=0, 1 if fl… | |
galoisisnormal |
galoisisnormal |
namespace | gal being as output by galoisinit, and subgrp a subgroup of gal as output by galoissubgroups, return 1 if subgrp is a no… | |
galoispermtopol |
galoispermtopol |
namespace | gal being a Galois group as output by galoisinit and perm a element of gal.group, return the polynomial defining the cor… | |
galoissplittinginit |
galoissplittinginit |
namespace | Galois group over Q of the splitting field of P, for P integral, monic and irreducible or given by a nf structure | |
galoissubcyclo |
galoissubcyclo |
namespace | computes a polynomial (in variable v) defining the subfield of Q(zeta_n) fixed by the subgroup H of (Z/nZ)* | |
galoissubfields |
galoissubfields |
namespace | output all the subfields of G; flag has the same meaning as for galoisfixedfield | |
galoissubgroups |
galoissubgroups |
namespace | output all the subgroups of G | |
gcharalgebraic |
gcharalgebraic |
namespace | returns a matrix whose columns form a basis of the algebraic Grossencharacters in gc | |
gcharconductor |
gcharconductor |
namespace | returns the conductor of chi, as a modulus over gc.bnf | |
gcharduallog |
gcharduallog |
namespace | returns logarithm vector of character chi in R^n | |
gchareval |
gchareval |
namespace | computes the evaluation chi(x) in C* if flag=1 and in C/Z if flag=0 | |
gcharidentify |
gcharidentify |
namespace | returns a Grossencharacter chi belonging to gc that approximately satisfies the constraints that chi_v is Lchiv[i] at th… | |
gcharinit |
gcharinit |
namespace | given a bnf as output by bnfinit and a modulus f, initializes data related to the group of Grossencharacters of conducto… | |
gcharisalgebraic |
gcharisalgebraic |
namespace | returns 1 if chi is an algebraic (type A0) character | |
gcharlocal |
gcharlocal |
namespace | if v is a place, return the local character chi_v | |
gcharlog |
gcharlog |
namespace | returns the internal representation (logarithm) of the ideal x suitable for computations in gc, as a column vector | |
gcharnewprec |
gcharnewprec |
namespace | given a Grossencharacter group , recomputes its invariants to ensure accurate results to current precision | |
idealadd |
idealadd |
namespace | sum of two ideals x and y in the number field defined by nf | |
idealaddtoone |
idealaddtoone |
namespace | if y is omitted, when the sum of the ideals in the number field K defined by nf and given in the vector x is equal to Z_… | |
idealappr |
idealappr |
namespace | x being a fractional ideal, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 … | |
idealchinese |
idealchinese |
namespace | x being a prime ideal factorization and y a vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all … | |
idealcoprime |
idealcoprime |
namespace | gives an element b in nf such that b | |
idealdiv |
idealdiv |
namespace | quotient x/y of two ideals x and y in HNF in the number field nf | |
idealdown |
idealdown |
namespace | finds the intersection of the ideal x with Q | |
idealfactor |
idealfactor |
namespace | factorization of the ideal x into prime ideals in the number field nf | |
idealfactorback |
idealfactorback |
namespace | given a factorization f, gives the ideal product back | |
idealfrobenius |
idealfrobenius |
namespace | returns the Frobenius element (pr|nf/Q) attached to the unramified prime ideal pr in prid format, in the Galois group ga… | |
idealhnf |
idealhnf |
namespace | hermite normal form of the ideal u in the number field nf if v is omitted | |
idealintersect |
idealintersect |
namespace | intersection of two ideals A and B in the number field defined by nf | |
idealinv |
idealinv |
namespace | inverse of the ideal x in the number field nf | |
idealismaximal |
idealismaximal |
namespace | if x is a maximal ideal, return it in prid form, else return 0 | |
idealispower |
idealispower |
namespace | return 1 if A = B^n is an n-th power else return 0 | |
ideallist |
ideallist |
namespace | vector of vectors L of all idealstar of all ideals of norm<=bound | |
ideallistarch |
ideallistarch |
namespace | list is a vector of vectors of bid’s as output by ideallist | |
ideallog |
ideallog |
namespace | if bid is a big ideal, as given by idealstar(nf,D,…), gives the vector of exponents on the generators bid.gen (even if… | |
idealmin |
idealmin |
namespace | pseudo-minimum of the ideal ix in the direction vdir in the number field nf | |
idealmul |
idealmul |
namespace | product of the two ideals x and y in the number field nf | |
idealnorm |
idealnorm |
namespace | norm of the ideal x in the number field nf | |
idealnumden |
idealnumden |
namespace | returns [A,B], where A,B are coprime integer ideals such that x = A/B | |
idealpow |
idealpow |
namespace | k-th power of the ideal x in HNF in the number field nf | |
idealprimedec |
idealprimedec |
namespace | ER2: NumberField.factor(p) |
prime ideal decomposition of the prime number p in the number field nf as a vector of prime ideals |
idealprincipalunits |
idealprincipalunits |
namespace | returns the structure [no, cyc, gen] of the multiplicative group (1 + pr) / (1 + pr^k) | |
idealramgroups |
idealramgroups |
namespace | let pr be a prime ideal in prid format, and gal the Galois group of the number field nf, return a vector g such that g[1… | |
idealred |
idealred |
namespace | LLL reduction of the ideal I in the number field nf along direction v, in HNF | |
idealredmodpower |
idealredmodpower |
namespace | return b such that x * b^n = v is small | |
idealstar |
idealstar |
namespace | gives the structure of (Z_K/N)^*, where N is a modulus (an ideal in any form or a vector [f0, foo], where f0 is an ideal… | |
idealtwoelt |
idealtwoelt |
namespace | two-element representation of an ideal x in the number field nf | |
idealval |
idealval |
namespace | valuation at pr given in idealprimedec format of the ideal x in the number field nf | |
matalgtobasis |
matalgtobasis |
namespace | nfalgtobasis applied to every element of the vector or matrix x | |
matbasistoalg |
matbasistoalg |
namespace | nfbasistoalg applied to every element of the matrix or vector x | |
modreverse |
modreverse |
namespace | reverse polmod of the polmod z, if it exists | |
newtonpoly |
newtonpoly |
namespace | Newton polygon of polynomial x with respect to the prime p | |
nfalgtobasis |
nfalgtobasis |
namespace | transforms the algebraic number x into a column vector on the integral basis nf.zk | |
nfbasis |
nfbasis |
namespace | integral basis of the field Q[a], where a is a root of the polynomial T, using the round 4 algorithm | |
nfbasistoalg |
nfbasistoalg |
namespace | transforms the column vector x on the integral basis into an algebraic number | |
nfcertify |
nfcertify |
namespace | returns a vector of composite integers used to certify nf.zk and nf.disc unconditionally (both are correct when the outp… | |
nfcompositum |
nfcompositum |
namespace | vector of all possible compositums of the number fields defined by the polynomials P and Q; flag is optional, whose bina… | |
nfdetint |
nfdetint |
namespace | multiple of the ideal determinant of the pseudo generating set x | |
nfdisc |
nfdisc |
namespace | discriminant of the number field defined by the polynomial T | |
nfdiscfactors |
nfdiscfactors |
namespace | [D, faD], where D = nfdisc(T), and faD is the factorization of |D| | |
nfeltadd |
nfeltadd |
namespace | element x+y in nf | |
nfeltdiv |
nfeltdiv |
namespace | element x/y in nf | |
nfeltdiveuc |
nfeltdiveuc |
namespace | gives algebraic integer q such that x-qy is small | |
nfeltdivmodpr |
nfeltdivmodpr |
namespace | this function is obsolete, use nfmodpr | |
nfeltdivrem |
nfeltdivrem |
namespace | gives [q,r] such that r=x-qy is small | |
nfeltembed |
nfeltembed |
namespace | complex embeddings of x at places given by vector pl | |
nfeltispower |
nfeltispower |
namespace | returns 1 if x is an n-th power in nf (and set y to an n-th root if present), else returns 0 | |
nfeltissquare |
nfeltissquare |
namespace | returns 1 if x is a square in nf (and sets y to a square root if present), else returns 0 | |
nfeltmod |
nfeltmod |
namespace | gives r such that r=x-qy is small with q algebraic integer | |
nfeltmul |
nfeltmul |
namespace | element x.y in nf | |
nfeltmulmodpr |
nfeltmulmodpr |
namespace | this function is obsolete, use nfmodpr | |
nfeltnorm |
nfeltnorm |
namespace | norm of x | |
nfeltpow |
nfeltpow |
namespace | element x^k in nf | |
nfeltpowmodpr |
nfeltpowmodpr |
namespace | this function is obsolete, use nfmodpr | |
nfeltreduce |
nfeltreduce |
namespace | gives r such that a-r is in the ideal id and r is small | |
nfeltreducemodpr |
nfeltreducemodpr |
namespace | this function is obsolete, use nfmodpr | |
nfeltsign |
nfeltsign |
namespace | signs of real embeddings of x at places given by vector pl | |
nfelttrace |
nfelttrace |
namespace | trace of x | |
nfeltval |
nfeltval |
namespace | valuation of element x at the prime pr as output by idealprimedec | |
nffactor |
nffactor |
namespace | factor polynomial T in number field nf | |
nffactorback |
nffactorback |
namespace | given a factorization f, returns the factored object back as an nf element | |
nffactormod |
nffactormod |
namespace | this routine is obsolete, use nfmodpr and factormod | |
nfgaloisapply |
nfgaloisapply |
namespace | apply the Galois automorphism aut to the object x (element or ideal) in the number field nf | |
nfgaloisconj |
nfgaloisconj |
namespace | list of conjugates of a root of the polynomial x=nf.pol in the same number field | |
nfgrunwaldwang |
nfgrunwaldwang |
namespace | a polynomial in the variable v defining a cyclic extension of nf (given in nf or bnf form) with local behavior prescribe… | |
nfhilbert |
nfhilbert |
namespace | if pr is omitted, global Hilbert symbol (a,b) in nf, that is 1 if X2-aY2-bZ^2 has a nontrivial solution (X,Y,Z) in nf,… | |
nfhnf |
nfhnf |
namespace | if x=[A,I], gives a pseudo-basis [B,J] of the module sum A_jI_j | |
nfhnfmod |
nfhnfmod |
namespace | if x=[A,I], and detx is a multiple of the ideal determinant of x, gives a pseudo-basis of the module sum A_jI_j | |
nfinit |
nfinit |
namespace | ER2: NumberField |
pol being a nonconstant irreducible polynomial in Q[X], returns an nf structure attached to the number field Q[X] / (pol… |
nfisideal |
nfisideal |
namespace | true(1) if x is an ideal in the number field nf, false(0) if not | |
nfisincl |
nfisincl |
namespace | let f and g define number fields, either irreducible rational polynomials or number fields as output by nfinit; tests wh… | |
nfisisom |
nfisisom |
namespace | as nfisincl but tests whether f is isomorphic to g | |
nfislocalpower |
nfislocalpower |
namespace | true(1) if a is an n-th power in the local field K_v, false(0) if not | |
nfkermodpr |
nfkermodpr |
namespace | this function is obsolete, use nfmodpr | |
nflist |
nflist |
namespace | finds number fields (up to isomorphism) with Galois group of Galois closure isomorphic to G, and s complex places | |
nfmodpr |
nfmodpr |
namespace | map x to the residue field mod pr | |
nfmodprinit |
nfmodprinit |
namespace | transform the prime ideal pr into modpr format necessary for all operations mod pr in the number field nf | |
nfmodprlift |
nfmodprlift |
namespace | lift x from residue field mod pr to nf | |
nfnewprec |
nfnewprec |
namespace | transform the number field data nf into new data using the current (usually larger) precision | |
nfpolsturm |
nfpolsturm |
namespace | number of distinct real roots of the polynomial s(T) where s runs through the real embeddings given by vector pl | |
nfresolvent |
nfresolvent |
namespace | In the case where the Galois closure of the number field defined by pol is S3, Dl, A4, S4, F5, A5, M21, or M42, gives th… | |
nfroots |
nfroots |
namespace | roots of polynomial x belonging to nf (Q if omitted) without multiplicity | |
nfrootsof1 |
nfrootsof1 |
namespace | number of roots of unity and primitive root of unity in the number field nf | |
nfsnf |
nfsnf |
namespace | if x=[A,I,J], outputs D=[d_1,…d_n] Smith normal form of x | |
nfsolvemodpr |
nfsolvemodpr |
namespace | this function is obsolete, use nfmodpr | |
nfsplitting |
nfsplitting |
namespace | defining polynomial S over Q for the splitting field of P, that is the smallest field over which P is totally split | |
nfsubfields |
nfsubfields |
namespace | finds all subfields of degree d of number field defined by pol (all subfields if d is null or omitted) | |
nfsubfieldscm |
nfsubfieldscm |
namespace | computes the maximal CM subfield of nf | |
nfsubfieldsmax |
nfsubfieldsmax |
namespace | computes the list of maximal subfields of nf | |
nfweilheight |
nfweilheight |
namespace | return the absolute Weil height of the vector v seen as an element of the projective space over the number field nf give… | |
polcompositum |
polcompositum |
namespace | vector of all possible compositums of the number fields defined by the polynomials P and Q; flag is optional, whose bina… | |
polgalois |
polgalois |
namespace | Galois group of the polynomial T (see manual for group coding) | |
polred |
polred |
namespace | deprecated, use polredbest | |
polredabs |
polredabs |
namespace | a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T… | |
polredbest |
polredbest |
namespace | reduction of the polynomial T (gives minimal polynomials only) | |
polredord |
polredord |
namespace | this function is obsolete, use polredbest | |
poltschirnhaus |
poltschirnhaus |
namespace | random Tschirnhausen transformation of the polynomial x | |
rnfalgtobasis |
rnfalgtobasis |
namespace | relative version of nfalgtobasis, where rnf is a relative numberfield | |
rnfbasis |
rnfbasis |
namespace | given a projective Z_K-module M as output by rnfpseudobasis or rnfsteinitz, gives either a basis of M if it is free, or … | |
rnfbasistoalg |
rnfbasistoalg |
namespace | relative version of nfbasistoalg, where rnf is a relative numberfield | |
rnfcharpoly |
rnfcharpoly |
namespace | characteristic polynomial of a over nf, where a belongs to the algebra defined by T over nf | |
rnfconductor |
rnfconductor |
namespace | conductor of the Abelian extension of bnf defined by T | |
rnfdedekind |
rnfdedekind |
namespace | relative Dedekind criterion over the number field K, represented by nf, applied to the order Z_K[X]/(P), modulo the prim… | |
rnfdet |
rnfdet |
namespace | given a pseudo-matrix M, compute its determinant | |
rnfdisc |
rnfdisc |
namespace | given a polynomial T with coefficients in nf, gives a 2-component vector [D,d], where D is the relative ideal discrimina… | |
rnfeltabstorel |
rnfeltabstorel |
namespace | transforms the element x from absolute to relative representation | |
rnfeltdown |
rnfeltdown |
namespace | expresses x on the base field if possible; returns an error otherwise | |
rnfeltnorm |
rnfeltnorm |
namespace | returns the relative norm N_{L/K}(x), as an element of K | |
rnfeltreltoabs |
rnfeltreltoabs |
namespace | transforms the element x from relative to absolute representation | |
rnfelttrace |
rnfelttrace |
namespace | returns the relative trace Tr_{L/K}(x), as an element of K | |
rnfeltup |
rnfeltup |
namespace | expresses x (belonging to the base field) on the relative field | |
rnfequation |
rnfequation |
namespace | given a pol with coefficients in nf, gives an absolute equation z of the number field defined by pol | |
rnfhnfbasis |
rnfhnfbasis |
namespace | given a bnf attached to a number field K and a projective Z_K module M given by a pseudo-matrix, returns either a true H… | |
rnfidealabstorel |
rnfidealabstorel |
namespace | transforms the ideal x from absolute to relative representation | |
rnfidealdown |
rnfidealdown |
namespace | finds the intersection of the ideal x with the base field | |
rnfidealfactor |
rnfidealfactor |
namespace | factor the ideal x into prime ideals in the number field nfinit(rnf) | |
rnfidealhnf |
rnfidealhnf |
namespace | relative version of idealhnf, where rnf is a relative numberfield | |
rnfidealmul |
rnfidealmul |
namespace | relative version of idealmul, where rnf is a relative numberfield | |
rnfidealnormabs |
rnfidealnormabs |
namespace | absolute norm of the ideal x | |
rnfidealnormrel |
rnfidealnormrel |
namespace | relative norm of the ideal x | |
rnfidealprimedec |
rnfidealprimedec |
namespace | return prime ideal decomposition of the maximal ideal pr of K in L/K; pr is also allowed to be a prime number p, in whic… | |
rnfidealreltoabs |
rnfidealreltoabs |
namespace | transforms the ideal x from relative to absolute representation | |
rnfidealtwoelt |
rnfidealtwoelt |
namespace | relative version of idealtwoelt, where rnf is a relative numberfield | |
rnfidealup |
rnfidealup |
namespace | lifts the ideal x (of the base field) to the relative field | |
rnfinit |
rnfinit |
namespace | T being an irreducible polynomial defined over the number field nf, initializes a vector of data necessary for working i… | |
rnfisabelian |
rnfisabelian |
namespace | T being a relative polynomial with coefficients in nf, return 1 if it defines an abelian extension, and 0 otherwise | |
rnfisfree |
rnfisfree |
namespace | given a bnf attached to a number field K and a projective Z_K module M given by a pseudo-matrix, return true (1) if M is… | |
rnfislocalcyclo |
rnfislocalcyclo |
namespace | true(1) if the l-extension attached to rnf is locally cyclotomic (locally contained in the Z_l extension of K_v at all p… | |
rnfisnorm |
rnfisnorm |
namespace | T is as output by rnfisnorminit applied to L/K | |
rnfisnorminit |
rnfisnorminit |
namespace | let K be defined by a root of pol, L/K the extension defined by polrel | |
rnfkummer |
rnfkummer |
namespace | this function is deprecated | |
rnflllgram |
rnflllgram |
namespace | given a pol with coefficients in nf and an order as output by rnfpseudobasis or similar, gives [[neworder],U], where new… | |
rnfnormgroup |
rnfnormgroup |
namespace | norm group (or Artin or Takagi group) corresponding to the Abelian extension of bnr.bnf defined by pol, where the module… | |
rnfpolred |
rnfpolred |
namespace | given a pol with coefficients in nf, finds a list of relative polynomials defining some subfields, hopefully simpler | |
rnfpolredabs |
rnfpolredabs |
namespace | given an irreducible pol with coefficients in nf, finds a canonical relative polynomial defining the same field | |
rnfpolredbest |
rnfpolredbest |
namespace | given a pol with coefficients in nf, finds a relative polynomial P defining the same field, hopefully simpler than pol; … | |
rnfpseudobasis |
rnfpseudobasis |
namespace | given an irreducible polynomial T with coefficients in nf, returns [A,J,D,d] where [A,J] is a pseudo basis of the maxima… | |
rnfsteinitz |
rnfsteinitz |
namespace | given a nf attached to a number field K and a projective module M given by a pseudo-matrix, returns [A,I,D,d] where (A,I… | |
subcyclohminus |
subcyclohminus |
namespace | Let F be the abelian number field contained in Q(zeta_f) corresponding to the subgroup H of (Z/fZ)^* | |
subcycloiwasawa |
subcycloiwasawa |
namespace | Let F be the abelian number field contained in Q(zeta_f) corresponding to the subgroup H of (Z/fZ)^* | |
subcyclopclgp |
subcyclopclgp |
namespace | Let F be the abelian number field contained in Q(zeta_f) corresponding to the subgroup H of (Z/fZ)^* | |
subgrouplist |
subgrouplist |
namespace | cyc being any object which has a ‘.cyc’ method giving the cyclic components for a finite Abelian group G, outputs the li… |
11. Associative algebras (68)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
algadd |
algadd |
namespace | element x+y in al (Hamilton quaternions if omitted) | |
algalgtobasis |
algalgtobasis |
namespace | transforms the element x of the algebra al into a column vector on the integral basis of al | |
algaut |
algaut |
namespace | the stored automorphism of the splitting field of the cyclic algebra al | |
algb |
algb |
namespace | the element b of the center of the cyclic algebra al used to define it | |
algbasis |
algbasis |
namespace | basis of the stored order of the central simple algebra al | |
algbasistoalg |
algbasistoalg |
namespace | transforms the column vector x on the integral basis of al into an element of al in algebraic form | |
algcenter |
algcenter |
namespace | center of the algebra al | |
algcentralproj |
algcentralproj |
namespace | projections of the algebra al on the orthogonal central idempotents z[i] | |
algchar |
algchar |
namespace | characteristic of the algebra al | |
algcharpoly |
algcharpoly |
namespace | (reduced) characteristic polynomial of b in al (Hamilton quaternions if omitted), with respect to the variable v | |
algdegree |
algdegree |
namespace | degree of the central simple algebra al | |
algdim |
algdim |
namespace | dimension of the algebra al | |
algdisc |
algdisc |
namespace | discriminant of the stored order of the algebra al | |
algdivl |
algdivl |
namespace | element xin al (Hamilton quaternions if omitted) | |
algdivr |
algdivr |
namespace | element x/y in al (Hamilton quaternions if omitted) | |
alggroup |
alggroup |
namespace | constructs the group algebra of gal over Q (resp | |
alggroupcenter |
alggroupcenter |
namespace | constructs the center of the group algebra of gal over Q (resp | |
alghasse |
alghasse |
namespace | the hasse invariant of the central simple algebra al at the place pl | |
alghassef |
alghassef |
namespace | the hasse invariant of the central simple algebra al at finite places | |
alghassei |
alghassei |
namespace | the hasse invariant of the central simple algebra al at infinite places | |
algindex |
algindex |
namespace | the index of the central simple algebra al | |
alginit |
alginit |
namespace | initializes the central simple algebra defined by data B, C | |
alginv |
alginv |
namespace | element 1/x in al (Hamilton quaternions if omitted) | |
alginvbasis |
alginvbasis |
namespace | basis of the natural order of the central simple algebra al in terms of the stored order | |
algisassociative |
algisassociative |
namespace | true (1) if the multiplication table mt is suitable for algtableinit(mt,p), false (0) otherwise | |
algiscommutative |
algiscommutative |
namespace | test whether the algebra al is commutative | |
algisdivision |
algisdivision |
namespace | tests whether the central simple algebra al is a division algebra | |
algisdivl |
algisdivl |
namespace | tests whether y is left divisible by x and sets z to the left quotient x | |
algisinv |
algisinv |
namespace | tests whether x is invertible in al (Hamilton quaternions if omitted) and sets ix to the inverse of x | |
algisramified |
algisramified |
namespace | tests whether the central simple algebra al is ramified, i.e | |
algissemisimple |
algissemisimple |
namespace | test whether the algebra al is semisimple | |
algissimple |
algissimple |
namespace | test whether the algebra al is simple | |
algissplit |
algissplit |
namespace | tests whether the central simple algebra al is split, i.e | |
alglatadd |
alglatadd |
namespace | the sum of the lattices lat1 and lat2 | |
alglatcontains |
alglatcontains |
namespace | tests whether the lattice lat contains the element x | |
alglatelement |
alglatelement |
namespace | returns the element of al whose coordinates on the Z-basis of lat are c | |
alglathnf |
alglathnf |
namespace | the lattice generated by the columns of m, assuming that this lattice contains d times the integral basis of al | |
alglatindex |
alglatindex |
namespace | the generalized index (lat2:lat1) | |
alglatinter |
alglatinter |
namespace | the intersection of the lattices lat1 and lat2 | |
alglatlefttransporter |
alglatlefttransporter |
namespace | the set of x in al such that x*lat1 is contained in lat2 | |
alglatmul |
alglatmul |
namespace | the lattice generated by the products of elements of lat1 and lat2 | |
alglatrighttransporter |
alglatrighttransporter |
namespace | the set of x in al such that lat1*x is contained in lat2 | |
alglatsubset |
alglatsubset |
namespace | tests whether lat1 is contained in lat2 and if true and ptindex is present, sets it to the index (lat2:lat1) | |
algmakeintegral |
algmakeintegral |
namespace | computes an integral multiplication table for an isomorphic algebra | |
algmul |
algmul |
namespace | element x*y in al (Hamilton quaternions if omitted) | |
algmultable |
algmultable |
namespace | multiplication table of al over its prime subfield | |
algneg |
algneg |
namespace | element -x in al (Hamilton quaternions if omitted) | |
algnorm |
algnorm |
namespace | (reduced) norm of x | |
algpoleval |
algpoleval |
namespace | T in K[X] evaluate T(b) in al (Hamilton quaternions if omitted) | |
algpow |
algpow |
namespace | element x^n in al (Hamilton quaternions if omitted) | |
algprimesubalg |
algprimesubalg |
namespace | prime subalgebra of the positive characteristic, semisimple algebra al | |
algquotient |
algquotient |
namespace | quotient of the algebra al by the two-sided ideal I | |
algradical |
algradical |
namespace | Jacobson radical of the algebra al | |
algramifiedplaces |
algramifiedplaces |
namespace | vector of the places of the center of al that ramify in al | |
algrandom |
algrandom |
namespace | random element in al (Hamilton quaternions if omitted) with coefficients in [-b,b] | |
algrelmultable |
algrelmultable |
namespace | multiplication table of the central simple algebra al over its center | |
algsimpledec |
algsimpledec |
namespace | [J,dec] where J is the Jacobson radical of al and dec is the decomposition into simple algebras of the semisimple algebr… | |
algsplit |
algsplit |
namespace | computes an isomorphism between al and M_d(F_q) | |
algsplittingdata |
algsplittingdata |
namespace | data stored in the central simple algebra al to compute a splitting of al over an extension | |
algsplittingfield |
algsplittingfield |
namespace | the stored splitting field of the central simple algebra al | |
algsqr |
algsqr |
namespace | element x^2 in al (Hamilton quaternions if omitted) | |
algsub |
algsub |
namespace | element x-y in al (Hamilton quaternions if omitted) | |
algsubalg |
algsubalg |
namespace | subalgebra of al with basis B | |
algtableinit |
algtableinit |
namespace | initializes the associative algebra over Q (resp | |
algtensor |
algtensor |
namespace | tensor product of al1 and al2 | |
algtomatrix |
algtomatrix |
namespace | left multiplication table of x (table algebra or abs=1) or image of x under a splitting of al (CSA and abs=0) | |
algtrace |
algtrace |
namespace | (reduced) trace of x | |
algtype |
algtype |
namespace | type of the algebra al |
12. Elliptic and hyperelliptic curves (103)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
ell2cover |
ell2cover |
namespace | if E is an elliptic curve over Q, returns a basis of the set of everywhere locally soluble 2-covers of the curve E | |
elladd |
elladd |
namespace | sum of the points z1 and z2 on elliptic curve E | |
ellak |
ellak |
namespace | computes the n-th Fourier coefficient of the L-function of the elliptic curve E (assumes E is an integral model) | |
ellan |
ellan |
namespace | computes the first n Fourier coefficients of the L-function of the elliptic curve E defined over a number field | |
ellanalyticrank |
ellanalyticrank |
namespace | returns the order of vanishing at s=1 of the L-function of the elliptic curve E and the value of the first nonzero deriv… | |
ellap |
ellap |
namespace | given an elliptic curve E defined over a finite field Fq, return the trace of Frobenius a_p = q+1-#E(Fq); for other fiel… | |
ellbil |
ellbil |
namespace | deprecated alias for ellheight(E,P,Q) | |
ellbsd |
ellbsd |
namespace | E being an elliptic curve over a number field, returns a real number c such that the BSD conjecture predicts that lfun(E… | |
ellcard |
ellcard |
namespace | given an elliptic curve E defined over a finite field Fq, return the order of the group E(Fq); for other fields of defin… | |
ellchangecurve |
ellchangecurve |
namespace | change data on elliptic curve according to v=[u,r,s,t] | |
ellchangepoint |
ellchangepoint |
namespace | change data on point or vector of points x on an elliptic curve according to v=[u,r,s,t] | |
ellchangepointinv |
ellchangepointinv |
namespace | change data on point or vector of points x on an elliptic curve according to v=[u,r,s,t], inverse of ellchangepoint | |
ellconvertname |
ellconvertname |
namespace | convert an elliptic curve name (as found in the elldata database) from a string to a triplet [conductor, isogeny class, … | |
elldivpol |
elldivpol |
namespace | n-division polynomial f_n for the curve E in the variable v | |
elleisnum |
elleisnum |
namespace | k being an even positive integer, computes the numerical value of the Eisenstein series of weight k at the lattice w, as… | |
elleta |
elleta |
namespace | w=[w1,w2], returns the vector [eta1,eta2] of quasi-periods attached to [w1,w2] | |
ellformaldifferential |
ellformaldifferential |
namespace | E elliptic curve, n integer | |
ellformalexp |
ellformalexp |
namespace | E elliptic curve, returns n terms of the formal elliptic exponential on E as a series in z | |
ellformallog |
ellformallog |
namespace | E elliptic curve, returns n terms of the elliptic logarithm as a series of t =-x/y | |
ellformalpoint |
ellformalpoint |
namespace | E elliptic curve, n integer; return the coordinates [x(t), y(t)] on the elliptic curve as a formal expansion in the form… | |
ellformalw |
ellformalw |
namespace | E elliptic curve, n integer; returns n terms of the formal expansion of w = -1/y in the formal parameter t = -x/y | |
ellfromeqn |
ellfromeqn |
namespace | given a genus 1 plane curve, defined by the affine equation f(x,y) = 0, return the coefficients [a1,a2,a3,a4,a6] of a We… | |
ellfromj |
ellfromj |
namespace | returns the coefficients [a1,a2,a3,a4,a6] of a fixed elliptic curve with j-invariant j | |
ellgenerators |
ellgenerators |
namespace | if E is an elliptic curve over the rationals, return the generators of the Mordell-Weil group attached to the curve | |
ellglobalred |
ellglobalred |
namespace | E being an elliptic curve over a number field, returns [N, v, c, faN, L], where N is the conductor of E, c is the produc… | |
ellgroup |
ellgroup |
namespace | given an elliptic curve E defined over a finite field Fq, returns the structure of the group E(Fq); for other fields of … | |
ellheegner |
ellheegner |
namespace | return a rational nontorsion point on the elliptic curve E assumed to be of rank 1 | |
ellheight |
ellheight |
namespace | Faltings height of the curve E, resp | |
ellheightmatrix |
ellheightmatrix |
namespace | gives the height matrix for vector of points x on elliptic curve E | |
ellidentify |
ellidentify |
namespace | look up the elliptic curve E in the elldata database and return [[N, M, …], C] where N is the name of the curve in Cre… | |
ellinit |
ellinit |
namespace | let x be a vector [a1,a2,a3,a4,a6], or [a4,a6] if a1=a2=a3=0, defining the curve Y^2 + a1.XY + a3.Y = X^3 + a2.X^2 + a4.… | |
ellintegralmodel |
ellintegralmodel |
namespace | given an elliptic curve E defined over a number field or Qp, returns an integral model | |
elliscm |
elliscm |
namespace | return 0 if the elliptic curve E, defined over a number field, is not CM, otherwise return the discriminant of its endom… | |
ellisdivisible |
ellisdivisible |
namespace | given E/K and P in E(K), checks whether P = [n]R for some R in E(K) and sets Q to one such R if so; the integer n >= 0 m… | |
ellisisom |
ellisisom |
namespace | return 0 if the elliptic curves E and F defined over the same number field are not isomorphic, otherwise return [u,r,s,t… | |
ellisogeny |
ellisogeny |
namespace | compute the image and isogeny corresponding to the quotient of E by the subgroup G | |
ellisogenyapply |
ellisogenyapply |
namespace | given an isogeny f and g either a point P (in the domain of f) or an isogeny, apply f to g: return the image of P under … | |
ellisomat |
ellisomat |
namespace | E being an elliptic curve over a number field K, returns a list of representatives of the isomorphism classes of ellipti… | |
ellisoncurve |
ellisoncurve |
namespace | true(1) if z is on elliptic curve E, false(0) if not | |
ellisotree |
ellisotree |
namespace | E being an elliptic curve over Q or a set of isogenous rational curves as given by ellisomat, return minimal models of t… | |
ellissupersingular |
ellissupersingular |
namespace | return 1 if the elliptic curve E, defined over a number field or a finite field, is supersingular at p, and 0 otherwise | |
ellj |
ellj |
namespace | elliptic j invariant of x | |
ellL1 |
ell_l1 |
namespace | renamed for PEP 8 | returns the value at s=1 of the derivative of order r of the L-function of the elliptic curve E |
elllocalred |
elllocalred |
namespace | E being an elliptic curve, returns [f,kod,[u,r,s,t],c], where f is the conductor’s exponent, kod is the Kodaira type for… | |
elllog |
elllog |
namespace | return the discrete logarithm of the point P of the elliptic curve E in base G | |
elllseries |
elllseries |
namespace | L-series at s of the elliptic curve E, where A a cut-off point close to 1 | |
ellmaninconstant |
ellmaninconstant |
namespace | let E be an elliptic curve over Q given by ellinit or a rational isogeny class given by ellisomat | |
ellminimaldisc |
ellminimaldisc |
namespace | E being an elliptic curve defined over a number field output by ellinit, return the minimal discriminant ideal of E | |
ellminimalmodel |
ellminimalmodel |
namespace | determines whether the elliptic curve E defined over a number field admits a global minimal model | |
ellminimaltwist |
ellminimaltwist |
namespace | E being an elliptic curve defined over Q, return a discriminant D such that the twist of E by D is minimal among all pos… | |
ellmoddegree |
ellmoddegree |
namespace | e being an elliptic curve defined over Q output by ellinit, compute the modular degree of e divided by the square of the… | |
ellmodulareqn |
ellmodulareqn |
namespace | given a prime N < 500, return a vector [P, t] where P(x,y) is a modular equation of level N | |
ellmul |
ellmul |
namespace | n times the point z on elliptic curve E (n in Z) | |
ellneg |
ellneg |
namespace | opposite of the point z on elliptic curve E | |
ellnonsingularmultiple |
ellnonsingularmultiple |
namespace | given E/Q and P in E(Q), returns the pair [R,n] where n is the least positive integer such that R = [n]P has everywhere … | |
ellorder |
ellorder |
namespace | order of the point z on the elliptic curve E over a number field or a finite field, 0 if nontorsion | |
ellordinate |
ellordinate |
namespace | y-coordinates corresponding to x-ordinate x on elliptic curve E | |
ellpadicbsd |
ellpadicbsd |
namespace | returns [r,Lp] where r is the (conjectural) analytic rank of the p-adic L-function attached to the quadratic twist E_D a… | |
ellpadicfrobenius |
ellpadicfrobenius |
namespace | matrix of the Frobenius at p>2 in the standard basis of H^1_dR(E) to absolute p-adic precision p^n | |
ellpadicheight |
ellpadicheight |
namespace | E elliptic curve/Q, P in E(Q), p prime, n an integer; returns the cyclotomic p-adic heights of P | |
ellpadicheightmatrix |
ellpadicheightmatrix |
namespace | gives the height-pairing matrix for vector of points Q on elliptic curve E | |
ellpadicL |
ellpadic_l |
namespace | renamed for PEP 8 | returns the value on a character of Z_{p}^* represented by an integer s or a vector [s1,s2] of the derivative of order r… |
ellpadiclambdamu |
ellpadiclambdamu |
namespace | returns the Iwasawa invariants for the p-adic L-function attached to E, twisted by (D,.) and the i-th power of the Teich… | |
ellpadiclog |
ellpadiclog |
namespace | returns the logarithm of P (in the kernel of reduction) to relative p-adic precision p^n | |
ellpadicregulator |
ellpadicregulator |
namespace | E elliptic curve/Q, S a vector of points in E(Q), p prime, n an integer; returns the p-adic cyclotomic regulator of the … | |
ellpadics2 |
ellpadics2 |
namespace | returns s2 to absolute p-adic precision p^n | |
ellperiods |
ellperiods |
namespace | w describes a complex period lattice ([w1,w2] or an ellinit structure) | |
ellpointtoz |
ellpointtoz |
namespace | lattice point z corresponding to the point P on the elliptic curve E | |
ellpow |
ellpow |
namespace | deprecated alias for ellmul | |
ellrank |
ellrank |
namespace | if E is an elliptic curve over Q, attempts to compute the Mordell-Weil group attached to the curve | |
ellrankinit |
ellrankinit |
namespace | if E is an elliptic curve over Q, initialize data for further calls to ellrank | |
ellratpoints |
ellratpoints |
namespace | E being an rational model of an elliptic curve, return a vector containing the affine rational points on the curve of na… | |
ellrootno |
ellrootno |
namespace | root number for the L-function of the elliptic curve E/Q at a prime p (including 0, for the infinite place); global root… | |
ellsaturation |
ellsaturation |
namespace | let E be an elliptic curve over Q and V be a vector of independent rational points on E of infinite order that generate … | |
ellsea |
ellsea |
namespace | computes the order of the group E(Fq) for the elliptic curve E, defined over a finite field, using SEA algorithm, with e… | |
ellsearch |
ellsearch |
namespace | returns all curves in the elldata database matching constraint N: given name (N = “11a1” or [11,0,1]), given isogeny cla… | |
ellsigma |
ellsigma |
namespace | computes the value at z of the Weierstrass sigma function attached to the lattice L, as given by ellperiods(,1) | |
ellsub |
ellsub |
namespace | difference of the points z1 and z2 on elliptic curve E | |
ellsupersingularj |
ellsupersingularj |
namespace | return a random supersingular j-invariant defined over F_p^2 if p is prime number, over the (finite) field of definition… | |
elltamagawa |
elltamagawa |
namespace | E being an elliptic curve over a number field, returns the global Tamagawa number of the curve | |
elltaniyama |
elltaniyama |
namespace | modular parametrization of elliptic curve E/Q | |
elltatepairing |
elltatepairing |
namespace | computes the Tate pairing of the two points P and Q on the elliptic curve E | |
elltors |
elltors |
namespace | torsion subgroup of elliptic curve E: order, structure, generators | |
elltrace |
elltrace |
namespace | sum of the Galois conjugates of the point P on elliptic curve E | |
elltwist |
elltwist |
namespace | returns an ell structure for the twist of the elliptic curve E by the quadratic extension defined by P (when P is a poly… | |
ellweilcurve |
ellweilcurve |
namespace | let E be an elliptic curve over Q given by ellinit or a rational isogeny class given by ellisomat | |
ellweilpairing |
ellweilpairing |
namespace | computes the Weil pairing of the two points of m-torsion P and Q on the elliptic curve E | |
ellwp |
ellwp |
namespace | computes the value at z of the Weierstrass P function attached to the lattice w, as given by ellperiods | |
ellxn |
ellxn |
namespace | return polynomials [A,B] in the variable v such that x([n]P) = (A/B)(t) for any P = [t,u] on E outside of n-torsion | |
ellzeta |
ellzeta |
namespace | computes the value at z of the Weierstrass Zeta function attached to the lattice w, as given by ellperiods(,1) | |
ellztopoint |
ellztopoint |
namespace | inverse of ellpointtoz | |
genus2igusa |
genus2igusa |
namespace | let PQ be a polynomial P, resp | |
genus2red |
genus2red |
namespace | let PQ be a polynomial P, resp | |
hyperellchangecurve |
hyperellchangecurve |
namespace | C being a nonsingular hyperelliptic model of a curve, apply the change of coordinate given by m | |
hyperellcharpoly |
hyperellcharpoly |
namespace | X being a nonsingular hyperelliptic curve defined over a finite field, return the characteristic polynomial of the Frobe… | |
hyperelldisc |
hyperelldisc |
namespace | X being a nonsingular hyperelliptic model of a curve, defined over a field of characteristic distinct from 2, returns it… | |
hyperellisoncurve |
hyperellisoncurve |
namespace | X being a nonsingular hyperelliptic model of a curve, test whether the point p is on the curve | |
hyperellminimaldisc |
hyperellminimaldisc |
namespace | C being a nonsingular integral hyperelliptic model of a curve, return the minimal discrminant of an integral model of C | |
hyperellminimalmodel |
hyperellminimalmodel |
namespace | C being a nonsingular integral hyperelliptic model of a curve, return an integral model of C with minimal discriminant | |
hyperellordinate |
hyperellordinate |
namespace | y-coordinates corresponding to x-ordinate x on hyperelliptic curve H | |
hyperellpadicfrobenius |
hyperellpadicfrobenius |
namespace | Q being a rational polynomial of degree d and X being the curve defined by y^2=Q(x), return the matrix of the Frobenius … | |
hyperellratpoints |
hyperellratpoints |
namespace | X being a nonsingular hyperelliptic curve given by an rational model, return a vector containing the affine rational poi… | |
hyperellred |
hyperellred |
namespace | C being a nonsingular integral hyperelliptic model of a curve, return an integral model of C with the same discriminant … |
13. L-functions (28)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
lfun |
lfun |
namespace | compute the L-function value L(s), or if D is set, the derivative of order D at s | |
lfunan |
lfunan |
namespace | compute the first n terms of the Dirichlet series attached to the L-function given by L (Lmath, Ldata or Linit) | |
lfunartin |
lfunartin |
namespace | returns the Ldata structure attached to the Artin L-function provided by the representation rho of the Galois group of t… | |
lfuncheckfeq |
lfuncheckfeq |
namespace | given an L-function (Lmath, Ldata or Linit), check whether the functional equation is satisfied | |
lfunconductor |
lfunconductor |
namespace | gives the conductor of the given L-function, expecting to find it in the interval [1,setN] | |
lfuncost |
lfuncost |
namespace | estimate the cost of running lfuninit(L,sdom,der) at current bit precision | |
lfuncreate |
lfuncreate |
namespace | given either an object such as a polynomial, elliptic curve, Dirichlet or Hecke character, eta quotient, etc., or an exp… | |
lfundiv |
lfundiv |
namespace | creates the Ldata structure (without initialization) corresponding to the quotient of the Dirichlet series given by L1 a… | |
lfundual |
lfundual |
namespace | creates the Ldata structure (without initialization) corresponding to the dual L-function of L | |
lfunetaquo |
lfunetaquo |
namespace | returns the Ldata structure attached to the modular form z->prod(i=1,#M[,1],eta(M[i,1]*z)^M[i,2]) | |
lfuneuler |
lfuneuler |
namespace | return the Euler factor at p of the L-function given by L (Lmath, Ldata or Linit) assuming the L-function admits an Eule… | |
lfungenus2 |
lfungenus2 |
namespace | returns the Ldata structure attached to the L-function attached to the genus-2 curve defined by y^2=F(x) or y^2+Q(x)*y=P… | |
lfunhardy |
lfunhardy |
namespace | variant of the Hardy L-function attached to L, used for plotting on the critical line | |
lfuninit |
lfuninit |
namespace | precompute data for evaluating the L-function given by ‘L’ (and its derivatives of order der, if set) in rectangular dom… | |
lfunlambda |
lfunlambda |
namespace | compute the completed L function Lambda(s), or if D is set, the derivative of order D at s | |
lfunmfspec |
lfunmfspec |
namespace | L corresponding to a modular eigenform, returns [ve,vo,om,op] in even weight, where ve (resp., vo) is the vector of even… | |
lfunmul |
lfunmul |
namespace | creates the Ldata structure (without initialization) corresponding to the product of the Dirichlet series given by L1 an… | |
lfunorderzero |
lfunorderzero |
namespace | computes the order of the possible zero of the L-function at the center k/2 of the critical strip | |
lfunparams |
lfunparams |
namespace | returns the parameters [N, k, vga] of the L-function defined by ldata (see lfuncreate) | |
lfunqf |
lfunqf |
namespace | returns the Ldata structure attached to the theta function of the lattice attached to the definite positive quadratic fo… | |
lfunrootres |
lfunrootres |
namespace | given the Ldata attached to an L-function (or the output of lfunthetainit), compute the root number and the residues | |
lfunshift |
lfunshift |
namespace | creates the Ldata structure (without initialization) corresponding to the function Ld such that Ld(s) = L(s-d) | |
lfunsympow |
lfunsympow |
namespace | returns the Ldata structure attached to the L-function attached to m-th symmetric power of the elliptic curve E defined … | |
lfuntheta |
lfuntheta |
namespace | compute the value of the m-th derivative at t of the theta function attached to the L-function given by data | |
lfunthetacost |
lfunthetacost |
namespace | estimates the cost of running lfunthetainit(L,tdom,m) at current bit precision | |
lfunthetainit |
lfunthetainit |
namespace | precompute data for evaluating the m-th derivative of theta functions with argument in domain tdom (by default t is real… | |
lfuntwist |
lfuntwist |
namespace | creates the Ldata structure (without initialization) corresponding to the twist of L by the primitive character attached… | |
lfunzeros |
lfunzeros |
namespace | lim being either an upper limit or a real interval, computes an ordered list of zeros of L(s) on the critical line up to… |
14. Hypergeometric motives (12)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
hgmalpha |
hgmalpha |
namespace | returns the alpha and beta parameters of the hypergeometric motive template H | |
hgmbydegree |
hgmbydegree |
namespace | outputs [L(0),…,L(n-1)] where L(w) is the list of cyclotomic parameters of all possible hypergeometric motive template… | |
hgmcoef |
hgmcoef |
namespace | (H,t) being a hypergeometric motive, returns the n-th coefficient of its L-function | |
hgmcoefs |
hgmcoefs |
namespace | (H,t) being a hypergeometric motive, returns the first n coefficients of its L-function, where Euler factors at wild pri… | |
hgmcyclo |
hgmcyclo |
namespace | returns the cyclotomic parameters (D,E) of the hypergeometric motive template H | |
hgmeulerfactor |
hgmeulerfactor |
namespace | (H,t) being a hypergeometric motive, returns the Euler factor P_p at the prime p; if present, set e to the valuation of … | |
hgmgamma |
hgmgamma |
namespace | returns the gamma vector of the hypergeometric motive template H | |
hgminit |
hgminit |
namespace | Create the template for a hypergeometric motive with parameters a and possibly b | |
hgmissymmetrical |
hgmissymmetrical |
namespace | is the hypergeometric motive template H symmetrical at t=1? | |
hgmparams |
hgmparams |
namespace | H being a hypergeometric motive template, returns [d, w, [P, T], M], where d is the degree, w the weight, P the Hodge po… | |
hgmtwist |
hgmtwist |
namespace | twist by 1/2 of alpha and beta of the hypergeometric motive template H | |
lfunhgm |
lfunhgm |
namespace | (H,t) being a hypergeometric motive, returns the corresponding lfuncreate data for use with the L function package |
15. Modular forms (69)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
lfunmf |
lfunmf |
namespace | If F is a modular form in mf, output the L-functions corresponding to its complex embeddings | |
mfatkin |
mfatkin |
namespace | Given an mfatk output by mfatk = mfatkininit(mf,Q) and a modular form f belonging to the space mf, returns the modular f… | |
mfatkineigenvalues |
mfatkineigenvalues |
namespace | given a modular form space mf and a primitive divisor Q of the level of mf, outputs the corresponding Atkin-Lehner eigen… | |
mfatkininit |
mfatkininit |
namespace | initializes data necessary for working with Atkin–Lehner operators W_Q, for now only the function mfatkin | |
mfbasis |
mfbasis |
namespace | If NK=[N,k,CHI] as in mfinit, gives a basis of the corresponding subspace of M_k(G_0(N),CHI) | |
mfbd |
mfbd |
namespace | F being a generalized modular form, return B(d)(F), where B(d) is the expanding operator tau -> d tau | |
mfbracket |
mfbracket |
namespace | compute the m-th Rankin-Cohen bracket of the generalized modular forms F and G | |
mfcoef |
mfcoef |
namespace | Compute the n-th Fourier coefficient a(n) of the generalized modular form F | |
mfcoefs |
mfcoefs |
namespace | Compute the vector of coefficients [a[0],a[d],…,a[nd]] of the modular form F | |
mfconductor |
mfconductor |
namespace | mf being output by mfinit and F a modular form, gives the smallest level at which F is defined | |
mfcosets |
mfcosets |
namespace | list of right cosets of G_0(N) i.e., matrices g_j in G such that G = U G_0(N) g_j | |
mfcuspisregular |
mfcuspisregular |
namespace | In the space defined by NK = [N,k,CHI] or NK = mf, determine if cusp in canonical format (oo or denominator dividing N) … | |
mfcusps |
mfcusps |
namespace | list of cusps of G_0(N) in the form a/b with b dividing N | |
mfcuspval |
mfcuspval |
namespace | valuation of modular form F in the space mf at cusp, which can be either oo or any rational number | |
mfcuspwidth |
mfcuspwidth |
namespace | width of cusp in Gamma_0(N) | |
mfDelta |
mf_delta |
namespace | renamed for PEP 8 | mf corresponding to the Ramanujan Delta function |
mfderiv |
mfderiv |
namespace | m-th formal derivative of the power series corresponding to the generalized modular form F, with respect to the differen… | |
mfderivE2 |
mfderiv_e2 |
namespace | renamed for PEP 8 | compute the Serre derivative (q.d/dq)F - kE_2F/12 of the generalized modular form F of weight k; and if m > 1, the m-th … |
mfdescribe |
mfdescribe |
namespace | gives a human-readable description of F, which is either a modular form space or a modular form | |
mfdim |
mfdim |
namespace | If NK=[N,k,CHI] as in mfinit, gives the dimension of the corresponding subspace of M_k(G_0(N),chi) | |
mfdiv |
mfdiv |
namespace | compute F/G for two modular forms F and G assuming that the quotient will not have poles at infinity | |
mfEH |
mf_eh |
namespace | renamed for PEP 8 | k>0 being in 1/2+Z, mf corresponding to the Cohen-Eisenstein series H_k of weight k on G_0(4) |
mfeigenbasis |
mfeigenbasis |
namespace | vector of the eigenforms for the space mf | |
mfeigensearch |
mfeigensearch |
namespace | search for normalized rational eigen cuspforms with quadratic characters given a few initial coefficients | |
mfeisenstein |
mfeisenstein |
namespace | create the Eisenstein E_k(CHI1,CHI2), where an omitted character is considered as trivial | |
mfEk |
mf_ek |
namespace | renamed for PEP 8 | mf corresponding to the standard Eisenstein series E_k for nonnegative even integer k |
mfembed |
mfembed |
namespace | if v is omitted, f must be a modular form or a modular form space with parameters [N,k,chi] and we return a vector of co… | |
mfeval |
mfeval |
namespace | computes the numerical value of the modular form F at the point vtau or the vector vtau of points in the completed upper… | |
mffields |
mffields |
namespace | If mf is output by mfinit, gives the vector of polynomials defining each Galois orbit of the new space | |
mffromell |
mffromell |
namespace | E being an elliptic curve defined over Q given by an integral model in ellinit format, computes a 3-component vector [mf… | |
mffrometaquo |
mffrometaquo |
namespace | modular form corresponding to the eta quotient matrix eta | |
mffromlfun |
mffromlfun |
namespace | L being an L-function representing a self-dual modular form, return [NK,space,v] where mf=mfinit(NK,space) contains the … | |
mffromqf |
mffromqf |
namespace | Q being an even positive definite quadratic form and P a homogeneous spherical polynomial for Q, computes a 3-component … | |
mfgaloisprojrep |
mfgaloisprojrep |
namespace | mf being an mf output by mfinit in weight 1, and F an eigenform, returns a polynomial defining the field fixed by the ke… | |
mfgaloistype |
mfgaloistype |
namespace | NK being either [N,1,CHI] or an mf output by mfinit in weight 1 , gives the vector of types of Galois representations at… | |
mfhecke |
mfhecke |
namespace | F being a modular form in space mf, returns T(n)F, where T(n) is the n-th Hecke operator | |
mfheckemat |
mfheckemat |
namespace | if vecn is an integer, matrix of the Hecke operator T(n) on the basis formed by mfbasis(mf), if it is a vector, vector o… | |
mfinit |
mfinit |
namespace | Create the space of modular forms corresponding to the data contained in NK and space | |
mfisCM |
mfis_cm |
namespace | renamed for PEP 8 | Tests whether the eigenform F is a CM form |
mfisequal |
mfisequal |
namespace | Checks whether the modular forms F and G are equal | |
mfisetaquo |
mfisetaquo |
namespace | if the generalized modular form f is a holomorphic eta quotient, return the eta quotient matrix, else return 0 | |
mfkohnenbasis |
mfkohnenbasis |
namespace | mf being a cuspidal space of half-integral weight k >= 3/2, gives a basis B of the Kohnen + space of mf as a matrix whos… | |
mfkohnenbijection |
mfkohnenbijection |
namespace | mf being a cuspidal space of half-integral weight returns [mf2,M,K,shi], where M is a matrix giving a Hecke-module isomo… | |
mfkohneneigenbasis |
mfkohneneigenbasis |
namespace | mf being a cuspidal space of half-integral weight k >= 3/2 and bij being the output of mfkohnenbijection(mf), outputs a … | |
mflinear |
mflinear |
namespace | vF being a vector of modular forms and v a vector of coefficients of same length, compute the linear combination of the … | |
mfmanin |
mfmanin |
namespace | Given the modular symbol FS associated to an eigenform F by mfsymbol(mf,F), computes the even and odd special polynomial… | |
mfmul |
mfmul |
namespace | Multiply the two forms F and G | |
mfnumcusps |
mfnumcusps |
namespace | number of cusps of Gamma_0(N) | |
mfparams |
mfparams |
namespace | If F is a modular form space, returns [N,k,CHI,space,Phi]: level, weight, character, and space code; where Phi is the cy… | |
mfperiodpol |
mfperiodpol |
namespace | period polynomial of the cuspidal part of the form f, in other words integral from 0 to ioo of (X-tau)^(k-2)f(tau) | |
mfperiodpolbasis |
mfperiodpolbasis |
namespace | basis of period polynomials for weight k | |
mfpetersson |
mfpetersson |
namespace | Petersson scalar product of the modular forms f and g belonging to the same modular form space mf, given by the correspo… | |
mfpow |
mfpow |
namespace | compute F^n | |
mfsearch |
mfsearch |
namespace | NK being of the form [N,k] with k possibly half-integral, search for a modular form with rational coefficients, of weigh… | |
mfshift |
mfshift |
namespace | Divide the form F by q^s omitting the remainder if there is one; s can be negative | |
mfshimura |
mfshimura |
namespace | F being a modular form of half-integral weight k >= 3/2 and D a positive squarefree integer, computes the Shimura lift G… | |
mfslashexpansion |
mfslashexpansion |
namespace | g being in M_2^+(Q), computes the Fourier expansion of f|_k g to n terms | |
mfspace |
mfspace |
namespace | identify the modular space mf, resp | |
mfsplit |
mfsplit |
namespace | mf containing the new space split the new space into Galois orbits of eigenforms of the newspace and return [vF,vK], whe… | |
mfsturm |
mfsturm |
namespace | Sturm bound for modular forms on G_0(N) and weight k, i.e., an upper bound for the order of the zero at infinity of a no… | |
mfsymbol |
mfsymbol |
namespace | Initialize data for working with all period polynomials of the modular form f: this is essential for efficiency for func… | |
mfsymboleval |
mfsymboleval |
namespace | evaluation of the modular symbol fs output by mfsymbol on the given path, where path is either a vector [s1,s2] or an in… | |
mftaylor |
mftaylor |
namespace | F being a modular form in M_k(SL_2(Z)), computes the first n+1 canonical Taylor expansion of F around tau=I | |
mfTheta |
mf_theta |
namespace | renamed for PEP 8 | the unary theta function corresponding to the primitive Dirichlet character psi, hence of weight 1/2 if psi is even, of … |
mftobasis |
mftobasis |
namespace | coefficients of the form F on the basis given by the mfbasis(mf) | |
mftocoset |
mftocoset |
namespace | M being a matrix in SL_2(Z) and Lcosets being mfcosets(N), find the right coset of G_0(N) to which M belongs | |
mftonew |
mftonew |
namespace | mf being a full or cuspidal space with parameters [N,k,chi] and F a cusp form in that space, returns a vector of 3-compo… | |
mftraceform |
mftraceform |
namespace | If NK=[N,k,CHI,.] as in mfinit with k integral, gives the trace form in the corresponding subspace of S_k(G_0(N),chi) | |
mftwist |
mftwist |
namespace | returns the twist of the form F by the integer D, i.e., the form G such that mfcoef(G,n)=(D/n)mfcoef(F,n), where (D/n) i… |
16. Modular symbols (31)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
msatkinlehner |
msatkinlehner |
namespace | M being a full modular symbol space of level N, as given by msinit, let Q | N, (Q,N/Q) = 1, and let H be a subspace stab… | |
mscosets |
mscosets |
namespace | gen being a system of generators for a group G and H being a subgroup of finite index of G, return a list of right coset… | |
mscuspidal |
mscuspidal |
namespace | M being a full modular symbol space, as given by msinit, return its cuspidal part S | |
msdim |
msdim |
namespace | M being a modular symbol space or subspace, return its dimension as a Q-vector space | |
mseisenstein |
mseisenstein |
namespace | M being a full modular symbol space, as given by msinit, return its Eisenstein subspace | |
mseval |
mseval |
namespace | M being a full modular symbol space, as given by msinit, s being a modular symbol from M and p being a path between two … | |
msfarey |
msfarey |
namespace | F being a Farey symbol attached to a group G contained in SL2(Z) and H a subgroup of G, return a Farey symbol attached t… | |
msfromcusp |
msfromcusp |
namespace | returns the modular symbol attached to the cusp c, where M is a modular symbol space of level N | |
msfromell |
msfromell |
namespace | return the [M, x], where M is msinit(N,2) and x is the modular symbol in M attached to the elliptic curve E/Q | |
msfromhecke |
msfromhecke |
namespace | given a msinit M and a vector v of pairs [p, P] (where p is prime and P is a polynomial with integer coefficients), retu… | |
msgetlevel |
msgetlevel |
namespace | M being a full modular symbol space, as given by msinit, return its level N | |
msgetsign |
msgetsign |
namespace | M being a full modular symbol space, as given by msinit, return its sign | |
msgetweight |
msgetweight |
namespace | M being a full modular symbol space, as given by msinit, return its weight k | |
mshecke |
mshecke |
namespace | M being a full modular symbol space, as given by msinit, p being a prime number, and H being a Hecke-stable subspace (M … | |
msinit |
msinit |
namespace | given G a finite index subgroup of SL(2,Z) and a finite dimensional representation V of GL(2,Q), creates a space of modu… | |
msissymbol |
msissymbol |
namespace | M being a full modular symbol space, as given by msinit, check whether s is a modular symbol attached to M | |
mslattice |
mslattice |
namespace | M being a full modular symbol space, as given by msinit, H a Q-subspace or a matrix of modular symbols | |
msnew |
msnew |
namespace | M being a full modular symbol space, as given by msinit, return its new cuspidal subspace | |
msomseval |
msomseval |
namespace | return the vectors of moments of the p-adic distribution attached to the path ‘path’ via the overconvergent modular symb… | |
mspadicinit |
mspadicinit |
namespace | M being a full modular symbol space, as given by msinit and a prime p, initialize technical data needed to compute with … | |
mspadicL |
mspadic_l |
namespace | renamed for PEP 8 | given mu from mspadicmoments (p-adic distributions attached to an overconvergent symbol PHI) returns the value on a char… |
mspadicmoments |
mspadicmoments |
namespace | given Mp from mspadicinit, an overconvergent eigensymbol PHI, and optionally a fundamental discriminant D coprime to p, … | |
mspadicseries |
mspadicseries |
namespace | given mu from mspadicmoments, returns the attached p-adic series with maximal p-adic precision, depending on the precisi… | |
mspathgens |
mspathgens |
namespace | M being a full modular symbol space, as given by msinit, return a set of Z[G]-generators for Div0(P1 Q) | |
mspathlog |
mspathlog |
namespace | M being a full modular symbol space, as given by msinit and p being a path between two elements in P^1(Q), return (p_i) … | |
mspetersson |
mspetersson |
namespace | M being a full modular symbol space, as given by msinit, calculate the intersection product {F,G} of modular symbols F a… | |
mspolygon |
mspolygon |
namespace | M describes a subgroup G of finite index in the modular group PSL2(Z), as given by msinit or a positive integer N (encod… | |
msqexpansion |
msqexpansion |
namespace | M being a full modular symbol space, as given by msinit, and projH being a projector on a Hecke-simple subspace, return … | |
mssplit |
mssplit |
namespace | M being a full modular symbol space, as given by msinit, and H being a subspace (the new subspace if omitted), split H i… | |
msstar |
msstar |
namespace | M being a full modular symbol space, as given by msinit, return the matrix of the * involution, induced by complex conju… | |
mstooms |
mstooms |
namespace | given Mp from mspadicinit, lift the (classical) eigen symbol phi to a distribution-valued overconvergent symbol in the s… |
17. Plotting (35)
| PARI | ER2 | Status | Note | Description |
|---|---|---|---|---|
parploth |
python | use Matplotlib | parallel version of ploth | |
parplothexport |
python | use Matplotlib | parallel version of plothexport | |
plot |
python | use Matplotlib | crude plot of expression expr, X goes from a to b, with Y ranging from Ymin to Ymax | |
plotarc |
python | use Matplotlib | if the cursor is at position (x1,y1), draws the ellipse that fits inside the box with diagonal (x1,y1) and (x2,y2) in re… | |
plotbox |
python | use Matplotlib | if the cursor is at position (x1,y1), draw a box with diagonal (x1,y1) and (x2,y2) in rectwindow w (cursor does not move… | |
plotclip |
python | use Matplotlib | clip the contents of the rectwindow to the bounding box (except strings) | |
plotcolor |
python | use Matplotlib | in rectwindow w, set default color to c | |
plotcopy |
python | use Matplotlib | copy the contents of rectwindow sourcew to rectwindow destw with offset (dx,dy) | |
plotcursor |
python | use Matplotlib | current position of cursor in rectwindow w | |
plotdraw |
python | use Matplotlib | draw rectwindow w | |
plotexport |
python | use Matplotlib | draw vector of rectwindows list as in plotdraw, returning the resulting picture as a character string; fmt is either “ps… | |
ploth |
python | use Matplotlib | plot of expression expr, X goes from a to b in high resolution | |
plothexport |
python | use Matplotlib | plot of expression expr, X goes from a to b in high resolution, returning the resulting picture as a character string wh… | |
plothraw |
python | use Matplotlib | plot in high resolution points whose x (resp | |
plothrawexport |
python | use Matplotlib | plot in high resolution points whose x (resp | |
plothsizes |
python | use Matplotlib | returns array of 8 elements: terminal width and height, sizes for ticks in horizontal and vertical directions, width and… | |
plotinit |
python | use Matplotlib | initialize rectwindow w to size x,y | |
plotkill |
python | use Matplotlib | erase the rectwindow w | |
plotlines |
python | use Matplotlib | draws an open polygon in rectwindow w where X and Y contain the x (resp | |
plotlinetype |
python | use Matplotlib | this function is obsolete; no graphing engine implement this functionality | |
plotmove |
python | use Matplotlib | move cursor to position x,y in rectwindow w | |
plotpoints |
python | use Matplotlib | draws in rectwindow w the points whose x (resp y) coordinates are in X (resp Y) | |
plotpointsize |
python | use Matplotlib | change the “size” of following points in rectwindow w | |
plotpointtype |
python | use Matplotlib | this function is obsolete; no graphing engine implement this functionality | |
plotrbox |
python | use Matplotlib | if the cursor is at (x1,y1), draw a box with diagonal (x1,y1)-(x1+dx,y1+dy) in rectwindow w (cursor does not move) | |
plotrecth |
python | use Matplotlib | writes to rectwindow w the curve output of ploth(w,X=a,b,expr,flag,n) | |
plotrecthraw |
python | use Matplotlib | plot graph(s) for data in rectwindow w, where data is a vector of vectors | |
plotrline |
python | use Matplotlib | if the cursor is at (x1,y1), draw a line from (x1,y1) to (x1+dx,y1+dy) (and move the cursor) in the rectwindow w | |
plotrmove |
python | use Matplotlib | move cursor to position (dx,dy) relative to the present position in the rectwindow w | |
plotrpoint |
python | use Matplotlib | draw a point (and move cursor) at position dx,dy relative to present position of the cursor in rectwindow w | |
plotscale |
python | use Matplotlib | scale the coordinates in rectwindow w so that x goes from x1 to x2 and y from y1 to y2 (y2<y1 is allowed) | |
plotstring |
python | use Matplotlib | draw in rectwindow w the string corresponding to x | |
psdraw |
python | use Matplotlib | obsolete function | |
psploth |
python | use Matplotlib | obsolete function | |
psplothraw |
python | use Matplotlib | obsolete function |