>
>
> iload "ex1-3.m";
Interactive-loading "ex1-3.m"
> //Modular polynomials
> p := 431;
> F<z>:= FiniteField(p, 2);
> E:= EllipticCurve([F!26, F!379]);
> E;
Elliptic Curve defined by y^2 = x^3 + 26*x + 379 over GF(431^2)
> IsSupersingular(E);
true
> j:= jInvariant(E);
> R<X,Y>:=PolynomialRing(F, 2);
> S<Z>:=PolynomialRing(F);
> Phi3 :=R!ClassicalModularPolynomial(3);
> f := Evaluate(Phi3, [j, Y]); f; Factorization(f);
Y^4 + 428*Y^3 + 267*Y^2 + 190*Y + 117
[
    <Y + z^37758, 1>,
    <Y + 281, 1>,
    <Y + z^112578, 1>,
    <Y + 324, 1>
]
> N := [f[1] : f in Roots(Evaluate(Phi3, [j, Z]))];
> N;
[ z^19698, 107, z^130638, 150 ]
> Phi16 := R!ClassicalModularPolynomial(16);
> f := Evaluate(Phi16, [j, Z]);
> Factorization(f);
[
    <Z, 2>,
    <Z + z^10275, 1>,
    <Z + z^37758, 1>,
    <Z + 75, 2>,
    <Z + 364, 1>,
    <Z + 242, 2>,
    <Z + 427, 1>,
    <Z + z^112578, 1>,
    <Z + 12, 2>,
    <Z + 197, 1>,
    <Z + 329, 2>,
    <Z + z^156045, 1>,
    <Z + 288, 2>,
    <Z + 412, 3>,
    <Z + 189, 2>
]
> Roots(f);
[ <0, 2>, <4, 1>, <z^19698, 1>, <419, 2>, <234, 1>, <102, 2>, <z^63165, 1>, <143, 2>, <19, 3>, <242, 2>, <z^103155, 1>,
<z^130638, 1>, <356, 2>, <67, 1>, <189, 2> ]
> primes := [ell : ell in  [1..11] | IsPrime(ell)];
> primes;
[ 2, 3, 5, 7, 11 ]
> for ell in primes do
for>   Phi := R!ClassicalModularPolynomial(ell);
for>   print ell, Roots(Evaluate(Phi, [Z,Z]));
for> end for;
2 [ <4, 1>, <242, 1>, <73, 2> ]
3 [ <0, 1>, <419, 2>, <125, 1>, <242, 2> ]
5 [ <4, 2>, <107, 2>, <419, 2>, <358, 1>, <19, 2>, <381, 1> ]
7 [ <0, 2>, <107, 2>, <241, 2>, <125, 2>, <61, 2>, <374, 1>, <356, 2>, <73, 1> ]
11 [ <112, 2>, <57, 2>, <107, 2>, <419, 1>, <274, 2>, <143, 2>, <242, 2>, <z^94686, 1>, <374, 2>, <z^128226, 1>, <422,
3>, <73, 2> ]
>
>
>
>