Elliptic Curve Automorphisms In Magma

148 Views Asked by At

I have the elliptic curve $E: y^2 = x^3+x$ over the field $\mathbb{F}_{43^2}$. I am trying to instantiate the automorphism $[i](x,y) = (-x,i*y)$, where $i^2=-1$ and the Frobenius Map $\pi(x,y) = (x^{43},y^{43})$. For my purposes I'd like to be able to instantiate them in standard form (i.e., $\phi(x,y) = (r(x),s(x)y)$ where $r,s\in F(x)$), so it would be nice if I could get the following code to work.

R<X>:=PolynomialRing(GF(43));
F<i>:=ext<GF(p)|X^2+1>;
E:=EllipticCurve([F|1,0]);
RR<x,y> = PolynomialRing(F,2);
iso:=Isogeny(E,E,1+0*x,42*x,i*y);

However, the bottom isogeny is the identity map.

iso;
    Elliptic curve isogeny from: CrvEll: E to CrvEll: E taking (x : y : 1) to (x : y : 1)

Similarly, trying to instantiate the Frobenius map in standard form also gives the identity; which I do as follows

pi:=Isogeny(E,E,1+0*x,x^43,((x^3+x)^21)*y);
pi;
    Elliptic curve isogeny from: CrvEll: E to CrvEll: E taking (x : y : 1) to (x : y : 1)