Gallery

A longer document, to show what the engine looks like in use. Everything here ran when the page was built.

Primes

p = nextprime(10^40)
factor(p - 1)
10000000000000000000000000000000000000121

[                      2 3]

[                      5 1]

[                     11 1]

[                     17 1]

[                  12973 1]

[             1821309023 1]

[56581485446137975519811 1]

p has 41 decimal digits and 133 binary digits.

Elliptic curves

The modular curve \(X_0(11)\):

E = ellinit([0, -1, 1, -10, -20]);
[E.disc, E.j]
[-161051, -122023936/161051]

Its conductor, analytic rank and the first coefficients of its \(L\)-series:

ellglobalred(E)[1]
ellanalyticrank(E)
ellan(E, 12)
11
[0, 0.25384186085591068433775892335090946104389844836612]
[1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2]
\\ y^2 + y = x^3 - x^2 - 10x - 20, solved for the upper y
up(x) = my(d = 1 + 4*(x^3 - x^2 - 10*x - 20)); if (d < 0, 0, (-1 + sqrt(d))/2);
plothexport("svg", X = 5.05, 12, up(X))
37.5035.24765.0512
Figure 1: The upper branch of the real locus of \(X_0(11)\)

Number fields

\(\mathbb{Q}(\sqrt[3]{2})\):

K = bnfinit(x^3 - 2, 1);
[K.disc, K.no, K.reg]
[-108, 1, 1.3473773483293841009181878914456530462830622733207]
K.zk
K.fu
[1, x, x^2]
[Mod(x - 1, x^3 - 2)]

High precision

precision: 50 applies to the whole page:

Pi
exp(1)
zeta(3)
3.1415926535897932384626433832795028841971693993751
2.7182818284590452353602874713526624977572470937000
1.2020569031595942853997381615114499907649862923405

Sums and series

sumdiv(2^10 * 3^5, d, d)
sum(n = 1, 1000, 1/n^2) * 1.0
Ser(exp(x), x, 8)
745108
1.6439345666815598031390580238222155896521034464937
1 + x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + 1/120*x^5 + 1/720*x^6 + 1/5040*x^7 + O(x^8)

Modular forms

mf = mfinit([1, 12], 1);
f = mfbasis(mf)[1];
mfcoefs(f, 10)
[0, 1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643, -115920]