1Function: ellissupersingular
2Section: elliptic_curves
3C-Name: ellissupersingular
4Prototype: iGDG
5Help: ellissupersingular(E,{p}): return 1 if the elliptic curve E, defined
6 over a number field or a finite field, is supersingular at p, and 0 otherwise.
7Doc:
8 Return 1 if the elliptic curve $E$ defined over a number field, $\Q_p$
9 or a finite field is supersingular at $p$, and $0$ otherwise.
10 If the curve is defined over a number field, $p$ must be explicitly given,
11 and must be a prime number, resp.~a maximal ideal, if the curve is defined
12 over $\Q$, resp.~a general number field: we return $1$ if and only if $E$
13 has supersingular good reduction at $p$.
14
15 Alternatively, $E$ can be given by its $j$-invariant in a finite field. In
16 this case $p$ must be omitted.
17 \bprog
18 ? setrand(1); \\ make the choice of g deterministic
19 ? g = ffprimroot(ffgen(7^5))
20 %1 = 4*x^4 + 5*x^3 + 6*x^2 + 5*x + 6
21 ? [g^n | n <- [1 .. 7^5 - 1], ellissupersingular(g^n)]
22 %2 = [6]
23
24 ? K = nfinit(y^3-2); P = idealprimedec(K, 2)[1];
25 ? E = ellinit([y,1], K);
26 ? ellissupersingular(E, P)
27 %5 = 1
28 ? Q = idealprimedec(K,5)[1];
29 ? ellissupersingular(E, Q)
30 %6 = 0
31 @eprog
32Variant: Also available is
33 \fun{int}{elljissupersingular}{GEN j} where $j$ is a $j$-invariant of a curve
34 over a finite field.
35