add ._scaling_factor() for .formal()[1]

Author: yyyyx4

src/sage/schemes/elliptic_curves/ell_curve_isogeny.py

@@ -2768,6 +2768,32 @@ def x_rational_map(self):
             self.__initialize_rational_maps()
         return self.__X_coord_rational_map
 
+    def _scaling_factor(self):
+        r"""
+        Return ``self.formal()[1]``, but faster.
+
+        EXAMPLES::
+
+            sage: E = EllipticCurve(GF(257^2), [0,1])
+            sage: phi = E.isogeny(E.lift_x(240))
+            sage: phi.degree()
+            43
+            sage: phi._scaling_factor()
+            1
+            sage: phi.dual()._scaling_factor()
+            43
+
+        ALGORITHM: The "inner" isogeny is normalized by construction,
+        so we only need to account for the scaling factors of a pre-
+        and post-isomorphism.
+        """
+        sc = Integer(1)
+        if self.__pre_isomorphism is not None:
+            sc *= self.__pre_isomorphism._scaling_factor()
+        if self.__post_isomorphism is not None:
+            sc *= self.__post_isomorphism._scaling_factor()
+        return sc
+
     def kernel_polynomial(self):
         r"""
         Return the kernel polynomial of this isogeny.

src/sage/schemes/elliptic_curves/hom_composite.py

@@ -765,6 +765,25 @@ def formal(self, prec=20):
             res = res(phi.formal(prec=prec))
         return res
 
+    def _scaling_factor(self):
+        r"""
+        Return ``self.formal()[1]``, but faster.
+
+        EXAMPLES::
+
+            sage: from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
+            sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
+            sage: E = EllipticCurve(GF(65537), [1,2,3,4,5])
+            sage: P = E.lift_x(7321)
+            sage: phi = EllipticCurveHom_composite(E, P)
+            sage: phi = WeierstrassIsomorphism(phi.codomain(), [7,8,9,10]) * phi
+            sage: phi.formal()
+            7*t + 65474*t^2 + 511*t^3 + 61316*t^4 + 20548*t^5 + 45511*t^6 + 37285*t^7 + 48414*t^8 + 9022*t^9 + 24025*t^10 + 35986*t^11 + 55397*t^12 + 25199*t^13 + 18744*t^14 + 46142*t^15 + 9078*t^16 + 18030*t^17 + 47599*t^18 + 12158*t^19 + 50630*t^20 + 56449*t^21 + 43320*t^22 + O(t^23)
+            sage: phi._scaling_factor()
+            7
+        """
+        return prod(phi._scaling_factor() for phi in self._phis)
+
     def is_injective(self):
         """
         Determine whether this composite morphism has trivial kernel.

src/sage/schemes/elliptic_curves/weierstrass_morphism.py

@@ -898,3 +898,16 @@ def __neg__(self):
         w = baseWI(-1, 0, -a1, -a3)
         urst = baseWI.__mul__(self, w).tuple()
         return WeierstrassIsomorphism(self._domain, urst, self._codomain)
+
+    def _scaling_factor(self):
+        r"""
+        Return ``self.formal()[1]``, but faster.
+
+        EXAMPLES::
+
+            sage: E = EllipticCurve(QQbar, [0,1])
+            sage: all(f._scaling_factor() == f.formal()[1] for f in E.automorphisms())
+            True
+        """
+        return self.u
+