> iload "ex1-2.m";
Interactive-loading "ex1-2.m"
> //Define the finite field
> p := 431;
> F<z> := FiniteField(p^2);
> E:= EllipticCurve([F!1, F!0]);
> //Compute the j-invariant
> jInvariant(E);
4
> // Point on an elliptic curve:
> P2:= E![0,0];
> // Check the order of this point
> Order(P2);
2
> // Compute an isogeny with kernel P:
> R<X> := PolynomialRing(F);
> // IsogenyFromKernel takes in the kernel polynomial of the isogeny
> E2, phi := IsogenyFromKernel(E, X - P2[1]);
> E2, phi;
Elliptic Curve defined by y^2 = x^3 + 427*x over GF(431^2)
Elliptic curve isogeny from: CrvEll: E to CrvEll: E2
taking (x : y : 1) to ((x^2 + 1) / x : (x^2*y + 430*y) / x^2 : 1)
> // Compute the dual isogeny
> DualIsogeny(phi);
Mapping from: CrvEll: E2 to CrvEll: E
Composition of Elliptic curve isogeny from: CrvEll: E2 to CrvEll: E
taking (x : y : 1) to ((323*x^2 + 1) / x : (z^132840*x^2*y + z^137160*y) / x^2 : 1) and
Mapping from: CrvEll: E to CrvEll: E
with equations :
430*$.1
z^139320*$.2
$.3
> // For an isogeny of prime degree, can use the following:
> function Isog(P)
function>   Edom := Curve(P);
function>   n := Order(P);
function> //  assert IsPrime(n);
function>   if n eq 2 then
function|if>     return IsogenyFromKernel(Edom, X - P[1]);
function|if>   else
function|if>   return IsogenyFromKernel(Edom, &*[X - (i*P)[1] : i in [1..(n-1) div 2]]);
function|if>   end if;
function> end function;
> // This code works for any degree
> function IsogenyFromPoint(P)
function>   Edom := Curve(P);
function>   n := Order(P);
function>   return IsogenyFromKernel(Edom, &*[X - (i*P)[1] : i in [1..n div 2]]);
function> end function;
> // Find a random point of order 16:
> P := 3^3*Random(E);
> while Order(P) ne 16 do
while>   P := 3^3*Random(E);
while> end while;
> // Find a second basis point for the E[16] torsion.
> Q := 3^3*Random(E);
> while (8*Q eq E!0) or (WeilPairing(P, Q, 16)^8 eq 1) do
while>   Q := 3^3*Random(E);
while> end while;
> // We can also check whether P, Q generate the full 16-torsion
> // By checking that 8P != 8Q.
> //while 8*P eq 8*Q and do
> //  Q := 3^3*Random(E);
> //end while;
> // Figuring out all the subgroups is not trivial (can't see it now).
> // generates all points of exact order 16
> points := [a*P+b*Q : a,b in [0..15] |  8*(a*P+b*Q) ne E!0] ;
> // I don't know how to define a list of isogenies from scratch
> // This forces a list of isogenies (Magma universe)
> Etemp, phitemp := IsogenyFromPoint(points[1]);
> Isog16 := [phitemp];
> Prune(~Isog16);
> for T in points do
for>   Etemp, phitemp := IsogenyFromPoint(T);
for>   if phitemp in Isog16 then
for|if>   else
for|if>     Isog16 := Isog16 cat [phitemp];
for|if>   end if;
for> end for;
> // Check we have all the isogenies
> #Isog16 eq 24;
true
>
>