Vélu isogeny formulas use incorrect a-invariants when pre-isomorphism is set

[yyyyx4]

With Sage 9.5.rc3:

sage: E = EllipticCurve(GF(71^2), [5,5])
sage: phi = E.isogeny(E.lift_x(0))
sage: E1 = E.change_weierstrass_model(9,8,7,6)
sage: phi = phi * E1.isomorphism_to(E)
sage: P = E1.lift_x(1)
sage: P in phi.domain()
True
sage: phi(P)
---------------------------------------------------------------------------

# ...

~/prg/sage/src/sage/schemes/elliptic_curves/ell_curve_isogeny.py in _call_(self, P)
   1182             outP = (tempX, tempY)
   1183
-> 1184         return self.__E2(outP)
   1185
   1186     def __getitem__(self, i):

# ...

TypeError: Coordinates [46, 3*z2 + 44, 1] do not define a point on Elliptic Curve defined by y^2 = x^3 + 5*x + 5 over Finite Field in z2 of size 71^2

The reason is the following code in EllipticCurveIsogeny.__compute_via_velu():

        a1 = self.__E1.a1()
        a3 = self.__E1.a3()

Here, .__E1 is the same as .domain(), but the Vélu formulas compute the "inner" isogeny. Thus, these should be the a1,a3 constants of the pre-isomorphism codomain.

CC: @JohnCremona @categorie @defeo

Component: elliptic curves

Author: Lorenz Panny

Branch/Commit: 3b823d0

Reviewer: John Cremona, Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/33214

[yyyyx4]

Author: Lorenz Panny

[yyyyx4]

Commit: 4408774

[yyyyx4]

Branch: public/fix_velu_isogeny_evaluation_with_pre_isomorphism

[yyyyx4]

New commits:

a1f4f7e	use correct a1,a3 constants when pre-isomorphism exists
4408774	add doctest for #33214

[fchapoton]

comment:2

looks good to me, and simple enough

John or Luca, do you approve ?

[JohnCremona]

comment:3

Well spotted, sounds very reasonable. Can you add something to the doctest which makes it clear that this is correct in the example, while the same code pre-patch is not? Some assertion which would fail before? Perhaps composing with a dual and checking that the composite multiplies by the degree? That way people can see that this is correct, without having to do some hard computations to verify.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

8f5da9e	Merge tag '9.6.beta6' into public/fix_velu_isogeny_evaluation_with_pre_isomorphism
3b823d0	add another doctest

[sagetrac-git]

Changed commit from 4408774 to 3b823d0

[yyyyx4]

comment:5

Thanks! I just added another test to verify that the rational maps of the isogeny with pre-isomorphism satisfy the (co)domain curve equations. (The rational maps are internally computed by evaluating the isogeny at a generic point, i.e., it uses the same code path that is corrected here.) Both tests fail with a current Sage and pass after this patch.

[JohnCremona]

Reviewer: John Cremona, Frédéric Chapoton

[JohnCremona]

comment:6

Replying to @yyyyx4:

Thanks! I just added another test to verify that the rational maps of the isogeny with pre-isomorphism satisfy the (co)domain curve equations. (The rational maps are internally computed by evaluating the isogeny at a generic point, i.e., it uses the same code path that is corrected here.) Both tests fail with a current Sage and pass after this patch.

That's great, just what I was suggesting.

[vbraun]

Changed branch from public/fix_velu_isogeny_evaluation_with_pre_isomorphism to 3b823d0