# Reductions of elliptic curves over number fields

I have found two elliptic curves over a number field whose reductions are isomorphic to supersingular elliptic curves (used Deuring Lifting Theorem). How can I check which reduction is isomorphic to which supersingular elliptic curve?

sage: H = hilbert_class_polynomial(-64)
sage: S.<l> = H.splitting_field()
sage: R = H.roots(ring = S)
sage: for r in R:
....:     j0 = r[0]
....:     E = EllipticCurve(j=j0)
....:     I = S.ideal(p)
....:     Ebar = E.reduction(I)
....:     Ebar
....:     Ebar.j_invariant()
Elliptic Curve defined by y^2 = x^3 + (67*lbar+26)*x + (54*lbar+71) over Residue field in lbar of Fractional ideal (179)
lbar
Elliptic Curve defined by y^2 = x^3 + (112*lbar+44)*x + (125*lbar+171) over Residue field in lbar of Fractional ideal (179)
178*lbar + 35

sage: p = 179
sage: F.<i> = GF(p^2, modulus=x^2+1)
sage: Rp = H.roots(ring=F)
sage: for r in Rp:
....:     j0 = r[0]
....:     EllipticCurve(j=j0)
....:     j0
Elliptic Curve defined by y^2 = x^3 + (10*i+35)*x + (155*i+121) over Finite Field in i of size 179^2
99*i + 107
Elliptic Curve defined by y^2 = x^3 + (169*i+35)*x + (24*i+121) over Finite Field in i of size 179^2
80*i + 107

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Can you please fix your code so that the example works? p is undefined on line 7.

( 2022-04-26 13:24:38 +0200 )edit

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Identifying the curves depends on an isomorphism between the residue field generated by lbar and the finite field generated by i. There's two possible isomorphisms, so with one choice the first curve on top is isomorphic to the first curve at the bottom, while with the other choice it's isomorphic to the second curve at the bottom. The fields are not canonically isomorphic, so there's no preferred isomorphism.

In general, support for isomorphisms of finite fields is not great in Sage. I recommend you factor polynomials and identify factors by hand.

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