Trac #32490: multiplication_by_m_isogeny is only correct up to automo…

…rphism

This is wrong:

{{{#!sage
sage: E = EllipticCurve(QQbar, [1,0])
sage: E.multiplication_by_m_isogeny(1).rational_maps()
(x, -y)
}}}

{{{#!sage
sage: E = EllipticCurve(GF(127), [1,0])
sage: P = E.random_point()
sage: E.multiplication_by_m_isogeny(5)(P)
(15 : 56 : 1)
sage: 5*P
(15 : 71 : 1)
sage: -5*P
(15 : 56 : 1)
sage: E.multiplication_by_m(5) ==
(-E.multiplication_by_m_isogeny(5)).rational_maps()
True
}}}

It isn't ''consistently'' wrong:

{{{#!sage
sage: E = EllipticCurve(QQ, [1,0])
sage: E.multiplication_by_m_isogeny(1).rational_maps()
(x, y)
}}}

----

The trouble is that the method first constructs **an** isogeny with the
correct kernel and codomain, then sets the precomputed rational maps.
However, the isomorphism to the prescribed codomain is not unique, and
picking the wrong one means the resulting isogeny is off by an
automorphism — most commonly `[-1]`.

The patch is not pretty, but it seems to fix the issue for now and
shouldn't cause any significant slowdowns.
Once (if) #32388 gets in, we should probably instead use an
`EllipticCurveHom` subclass specifically for scalar multiplications that
imitates a normal isogeny but with different internals. I do have a
draft of this ready and it's not only much cleaner, but also resolves
part of #8014.

URL: https://trac.sagemath.org/32490
Reported by: lorenz
Ticket author(s): Lorenz Panny
Reviewer(s): Travis Scrimshaw

src/sage/schemes/elliptic_curves/ell_generic.py

@@ -2150,7 +2150,7 @@ def multiplication_by_m(self, m, x_only=False):
     def multiplication_by_m_isogeny(self, m):
         r"""
         Return the ``EllipticCurveIsogeny`` object associated to the
-        multiplication-by-`m` map on self.
+        multiplication-by-`m` map on this elliptic curve.
 
         The resulting isogeny will
         have the associated rational maps (i.e. those returned by
@@ -2169,22 +2169,51 @@ def multiplication_by_m_isogeny(self, m):
         OUTPUT:
 
         - An ``EllipticCurveIsogeny`` object associated to the
-          multiplication-by-`m` map on self.
+          multiplication-by-`m` map on this elliptic curve.
 
         EXAMPLES::
 
             sage: E = EllipticCurve('11a1')
             sage: E.multiplication_by_m_isogeny(7)
             Isogeny of degree 49 from Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
 
+        TESTS:
+
+        Tests for :trac:`32490`::
+
+            sage: E = EllipticCurve(QQbar, [1,0])
+            sage: E.multiplication_by_m_isogeny(1).rational_maps()
+            (x, y)
+
+        ::
+
+            sage: E = EllipticCurve_from_j(GF(31337).random_element())
+            sage: P = E.random_point()
+            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: 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
+            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))
         """
         mx, my = self.multiplication_by_m(m)
 
         torsion_poly = self.torsion_polynomial(m).monic()
         phi = self.isogeny(torsion_poly, codomain=self)
         phi._EllipticCurveIsogeny__initialize_rational_maps(precomputed_maps=(mx, my))
 
-        return phi
+        # trac 32490: using codomain=self can give a wrong isomorphism
+        for aut in self.automorphisms():
+            psi = aut * phi
+            if psi.rational_maps() == (mx, my):
+                return psi
+
+        assert False, 'bug in multiplication_by_m_isogeny()'
 
     def isomorphism_to(self, other):
         """