gh-37415: make EllipticCurvePoint_field inherit from EllipticCurvePoint

    
The method `EllipticCurve_generic.__contains__()` checks for inheritance
from `EllipticCurvePoint`; however, the other classes for points on
elliptic curves do not inherit from this class.  This pull request fixes
this. Note that there is some overlap with #33228.

This change makes `P in E` much faster:
- Over **Q**:
```python
E = EllipticCurve('37a1')
P = E((0, -1))
%timeit P in E
```
Before: `35.3 µs ± 186 ns per loop (mean ± std. dev. of 7 runs, 10,000
loops each)`
After: `324 ns ± 0.535 ns per loop (mean ± std. dev. of 7 runs,
1,000,000 loops each)`

- Over a finite field:
```python
F.<a> = FiniteField((23, 2), modulus='random')
E = EllipticCurve_from_j(F.random_element())
P = E.random_point()
%timeit P in E
```
Before: `36.4 µs ± 542 ns per loop (mean ± std. dev. of 7 runs, 10,000
loops each)`
After: `333 ns ± 5.03 ns per loop (mean ± std. dev. of 7 runs, 1,000,000
loops each)`
    
URL: #37415
Reported by: Peter Bruin
Reviewer(s): Kwankyu Lee, Peter Bruin

src/sage/schemes/elliptic_curves/ell_generic.py

@@ -456,10 +456,7 @@ def __contains__(self, P):
                 P = self(P)
             except TypeError:
                 return False
-        if P.curve() == self:
-            return True
-        x, y, a = P[0], P[1], self.ainvs()
-        return y**2 + a[0]*x*y + a[2]*y == x**3 + a[1]*x**2 + a[3]*x + a[4]
+        return P.curve() == self
 
     def __call__(self, *args, **kwds):
         r"""
@@ -572,10 +569,7 @@ def __call__(self, *args, **kwds):
         P = args[0]
         if isinstance(P, groups.AdditiveAbelianGroupElement) and isinstance(P.parent(),ell_torsion.EllipticCurveTorsionSubgroup):
             return self(P.element())
-        if isinstance(args[0],
-                      (ell_point.EllipticCurvePoint_field,
-                       ell_point.EllipticCurvePoint_number_field,
-                       ell_point.EllipticCurvePoint)):
+        if isinstance(args[0], ell_point.EllipticCurvePoint):
             if P.curve() is self:
                 return P
             # check if denominator of the point contains a factor of the

src/sage/schemes/elliptic_curves/ell_point.py

@@ -1,18 +1,15 @@
 r"""
 Points on elliptic curves
 
-The base class ``EllipticCurvePoint_field``, derived from
-``AdditiveGroupElement``, provides support for points on elliptic
-curves defined over general fields.  The derived classes
-``EllipticCurvePoint_number_field`` and
-``EllipticCurvePoint_finite_field`` provide further support for point
-on curves defined over number fields (including the rational field
+The base class :class:`EllipticCurvePoint` currently provides little
+functionality of its own.  Its derived class
+:class:`EllipticCurvePoint_field` provides support for points on
+elliptic curves over general fields.  The derived classes
+:class:`EllipticCurvePoint_number_field` and
+:class:`EllipticCurvePoint_finite_field` provide further support for
+points on curves over number fields (including the rational field
 `\QQ`) and over finite fields.
 
-The class ``EllipticCurvePoint``, which is based on
-``SchemeMorphism_point_projective_ring``, currently has little extra
-functionality.
-
 EXAMPLES:
 
 An example over `\QQ`::
@@ -107,7 +104,6 @@
 
 - Mariah Lenox (March 2011) -- Added ``tate_pairing`` and ``ate_pairing``
   functions to ``EllipticCurvePoint_finite_field`` class
-
 """
 
 # ****************************************************************************
@@ -134,6 +130,8 @@
 from sage.rings.real_mpfr import RealField
 from sage.rings.real_mpfr import RR
 import sage.groups.generic as generic
+
+from sage.structure.element import AdditiveGroupElement
 from sage.structure.sequence import Sequence
 from sage.structure.richcmp import richcmp
 
@@ -153,14 +151,41 @@
     PariError = ()
 
 
-class EllipticCurvePoint(SchemeMorphism_point_projective_ring):
+class EllipticCurvePoint(AdditiveGroupElement,
+                         SchemeMorphism_point_projective_ring):
     """
     A point on an elliptic curve.
     """
-    pass
+    def curve(self):
+        """
+        Return the curve that this point is on.
+
+        This is a synonym for :meth:`scheme`.
 
+        EXAMPLES::
 
-class EllipticCurvePoint_field(SchemeMorphism_point_abelian_variety_field):
+            sage: E = EllipticCurve('389a')
+            sage: P = E([-1, 1])
+            sage: P.curve()
+            Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
+
+            sage: E = EllipticCurve(QQ, [1, 1])
+            sage: P = E(0, 1)
+            sage: P.scheme()
+            Elliptic Curve defined by y^2 = x^3 + x + 1 over Rational Field
+            sage: P.scheme() == P.curve()
+            True
+            sage: x = polygen(ZZ, 'x')
+            sage: K.<a> = NumberField(x^2 - 3,'a')                                      # needs sage.rings.number_field
+            sage: P = E.base_extend(K)(1, a)                                            # needs sage.rings.number_field
+            sage: P.scheme()                                                            # needs sage.rings.number_field
+            Elliptic Curve defined by y^2 = x^3 + x + 1 over Number Field in a with defining polynomial x^2 - 3
+        """
+        return self.scheme()
+
+
+class EllipticCurvePoint_field(EllipticCurvePoint,
+                               SchemeMorphism_point_abelian_variety_field):
     """
     A point on an elliptic curve over a field.  The point has coordinates
     in the base field.
@@ -275,7 +300,6 @@ def __init__(self, curve, v, check=True):
             v = (R.zero(), R.one(), R.zero())
 
         SchemeMorphism_point_abelian_variety_field.__init__(self, point_homset, v, check=check)
-        # AdditiveGroupElement.__init__(self, point_homset)
 
         self.normalize_coordinates()
 
@@ -426,39 +450,6 @@ def __pari__(self):
         else:
             return pari([0])
 
-    def scheme(self):
-        """
-        Return the scheme of this point, i.e., the curve it is on.
-        This is synonymous with :meth:`curve` which is perhaps more
-        intuitive.
-
-        EXAMPLES::
-
-            sage: E = EllipticCurve(QQ,[1,1])
-            sage: P = E(0,1)
-            sage: P.scheme()
-            Elliptic Curve defined by y^2 = x^3 + x + 1 over Rational Field
-            sage: P.scheme() == P.curve()
-            True
-            sage: x = polygen(ZZ, 'x')
-            sage: K.<a> = NumberField(x^2 - 3,'a')                                      # needs sage.rings.number_field
-            sage: P = E.base_extend(K)(1, a)                                            # needs sage.rings.number_field
-            sage: P.scheme()                                                            # needs sage.rings.number_field
-            Elliptic Curve defined by y^2 = x^3 + x + 1
-            over Number Field in a with defining polynomial x^2 - 3
-        """
-        # The following text is just not true: it applies to the class
-        # EllipticCurvePoint, which appears to be never used, but does
-        # not apply to EllipticCurvePoint_field which is simply derived
-        # from AdditiveGroupElement.
-        #
-        # "Technically, points on curves in Sage are scheme maps from
-        #  the domain Spec(F) where F is the base field of the curve to
-        #  the codomain which is the curve.  See also domain() and
-        #  codomain()."
-
-        return self.codomain()
-
     def order(self):
         r"""
         Return the order of this point on the elliptic curve.
@@ -497,19 +488,6 @@ def order(self):
 
     additive_order = order
 
-    def curve(self):
-        """
-        Return the curve that this point is on.
-
-        EXAMPLES::
-
-            sage: E = EllipticCurve('389a')
-            sage: P = E([-1,1])
-            sage: P.curve()
-            Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
-        """
-        return self.scheme()
-
     def __bool__(self):
         """
         Return ``True`` if this is not the zero point on the curve.