make WeierstrassIsomorphism behave (more) like EllipticCurveIsogeny

[yyyyx4]

This ticket is the obvious next step after #32388: We move lots of code from EllipticCurveIsogeny to EllipticCurveHom, and implement the missing EllipticCurveHom methods in WeierstrassIsomorphism.

This should make WeierstrassIsomorphism close to indistinguishable from EllipticCurveIsogeny.

All the really new code is in weierstrass_morphism.py and should be straightforward. The other changes are just moving things around and some small tweaks that shouldn't cause any observable changes in behaviour.

(The only "real" dependency is #32388. I've merged the others into this branch too because it made sense, but in principle this could be rebased on top of #32388 alone if one of the others gets stalled for some reason.)

Depends on #32388
Depends on #32430
Depends on #32495
Depends on #32490

CC: @defeo @JohnCremona

Component: elliptic curves

Keywords: isogenies, refactoring

Author: Lorenz Panny

Branch/Commit: 444330c

Reviewer: John Cremona

Issue created by migration from https://trac.sagemath.org/ticket/32502

[sagetrac-git]

Changed commit from 8414b20 to 440cff0

[sagetrac-git]

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

9161ffa	make WeierstrassIsomorphism behave like EllipticCurveIsogeny
440cff0	add test for comparisons between different EllipticCurveHom children

[yyyyx4]

comment:2

See here for a diff without all the dependencies:

• sagemath/sagetrac-mirror@80b0bb11171daf79122859a83d0940bd3fd2423a&id2=8414b20850b468d0b498bab5ac0e2167f97052f5

[sagetrac-git]

Changed commit from 440cff0 to 80b0bb1

[sagetrac-git]

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

80b0bb1

reflect this change in the documentation

[sagetrac-git]

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

894b145

Merge tag '9.5.beta3' into public/make_WeierstrassIsomorphism_behave_like_EllipticCurveIsogeny

[sagetrac-git]

Changed commit from 80b0bb1 to 894b145

[slel]

comment:5

Sorry, I cannot review in depth; just a few things.

Some .. NOTE:: blocks are indented by only three spaces
instead of the required four. Please fix.

Possible cosmetic changes (feel free to ignore):

-            sage: [E.multiplication_by_m_isogeny(m)(P) == m*P for m in (1,2,3,5,7,9)]
-            [True, True, True, True, True, True]
+            sage: all(E.multiplication_by_m_isogeny(m)(P) == m*P
+            ....:     for m in (1, 2, 3, 5, 7, 9))
+            True

             sage: E = EllipticCurve('99.a1')
             sage: E.multiplication_by_m_isogeny(5)
-            Isogeny of degree 25 from Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 17*x + 30 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 17*x + 30 over Rational Field
+            Isogeny of degree 25
+              from Elliptic Curve
+                defined by y^2 + x*y + y = x^3 - x^2 - 17*x + 30
+                over Rational Field
+              to Elliptic Curve
+                defined by y^2 + x*y + y = x^3 - x^2 - 17*x + 30
+                over Rational Field
             sage: E.multiplication_by_m_isogeny(2).rational_maps()
             ((1/4*x^4 + 33/4*x^2 - 121/2*x + 363/4)/(x^3 - 3/4*x^2 - 33/2*x + 121/4),
-             (-1/256*x^7 + 1/128*x^6*y - 7/256*x^6 - 3/256*x^5*y - 105/256*x^5 - 165/256*x^4*y + 1255/256*x^4 + 605/128*x^3*y - 473/64*x^3 - 1815/128*x^2*y - 10527/256*x^2 + 2541/128*x*y + 4477/32*x - 1331/128*y - 30613/256)/(1/16*x^6 - 3/32*x^5 - 519/256*x^4 + 341/64*x^3 + 1815/128*x^2 - 3993/64*x + 14641/256))
+             (-1/256*x^7 + 1/128*x^6*y - 7/256*x^6 - 3/256*x^5*y - 105/256*x^5
+              - 165/256*x^4*y + 1255/256*x^4 + 605/128*x^3*y - 473/64*x^3
+              - 1815/128*x^2*y - 10527/256*x^2 + 2541/128*x*y + 4477/32*x
+              - 1331/128*y - 30613/256)/(1/16*x^6 - 3/32*x^5 - 519/256*x^4
+              + 341/64*x^3 + 1815/128*x^2 - 3993/64*x + 14641/256))

-            sage: [a == b for a in (wE,mE) for b in (wF,mF)]
-            [False, False, False, False]
+            sage: any(a == b for a in (wE, mE) for b in (wF, mF))
+            False

-        #TODO: could have a default implementation that simply
-        #      returns the first component of rational_maps()
+        # TODO: could have a default implementation that simply
+        # returns the first component of rational_maps()

             sage: psi = E1.isogeny(iso.kernel_polynomial(), codomain=E2); psi
-            Isogeny of degree 1 from Elliptic Curve defined by y^2 + 11*x*y + 33*y = x^3 + 22*x^2 + 44*x + 55 over Rational Field to Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 684*x + 6681 over Rational Field
+            Isogeny of degree 1
+              from Elliptic Curve
+                defined by y^2 + 11*x*y + 33*y = x^3 + 22*x^2 + 44*x + 55
+                over Rational Field
+              to Elliptic Curve
+                defined by y^2 + x*y = x^3 + x^2 - 684*x + 6681
+                over Rational Field

[slel]

Reviewer: Samuel Lelièvre, ...