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