sums of elliptic-curve morphisms

src/doc/en/reference/arithmetic_curves/index.rst

@@ -18,10 +18,11 @@ Maps between them
    :maxdepth: 1
 
    sage/schemes/elliptic_curves/hom
+   sage/schemes/elliptic_curves/hom_composite
+   sage/schemes/elliptic_curves/hom_sum
    sage/schemes/elliptic_curves/weierstrass_morphism
    sage/schemes/elliptic_curves/ell_curve_isogeny
    sage/schemes/elliptic_curves/hom_velusqrt
-   sage/schemes/elliptic_curves/hom_composite
    sage/schemes/elliptic_curves/hom_scalar
    sage/schemes/elliptic_curves/hom_frobenius
    sage/schemes/elliptic_curves/isogeny_small_degree

src/sage/schemes/elliptic_curves/ell_curve_isogeny.py

@@ -2879,28 +2879,19 @@ def kernel_polynomial(self):
             self.__init_kernel_polynomial()
         return self.__kernel_polynomial
 
-    def is_separable(self):
+    def inseparable_degree(self):
         r"""
-        Determine whether or not this isogeny is separable.
+        Return the inseparable degree of this isogeny.
 
-        Since :class:`EllipticCurveIsogeny` only implements
-        separable isogenies, this method always returns ``True``.
+        Since this class only implements separable isogenies,
+        this method always returns one.
 
-        EXAMPLES::
-
-            sage: E = EllipticCurve(GF(17), [0,0,0,3,0])
-            sage: phi = EllipticCurveIsogeny(E,  E((0,0)))
-            sage: phi.is_separable()
-            True
-
-        ::
+        TESTS::
 
-            sage: E = EllipticCurve('11a1')
-            sage: phi = EllipticCurveIsogeny(E, E.torsion_points())
-            sage: phi.is_separable()
-            True
+            sage: EllipticCurveIsogeny.inseparable_degree(None)
+            1
         """
-        return True
+        return Integer(1)
 
     def _set_pre_isomorphism(self, preWI):
         """

src/sage/schemes/elliptic_curves/ell_field.py

@@ -2049,13 +2049,13 @@
 
     raise NotImplementedError(f'cannot compute {name} model')
 
-def point_of_order(E, l):
+def point_of_order(E, n):
     r"""
     Given an elliptic curve `E` over a finite field or a number field
-    and an integer `\ell \geq 1`, construct a point of order `\ell` on `E`,
+    and an integer `n \geq 1`, construct a point of order `n` on `E`,
     possibly defined over an extension of the base field of `E`.
 
-    Currently only prime values of `\ell` are supported.
+    Currently only prime powers `n` are supported.
 
     EXAMPLES::
 
@@ -2070,6 +2070,13 @@
         sage: P.curve().a_invariants()
         (1, 2, 3, 4, 5)
 
+    ::
+
+        sage: Q = point_of_order(E, 8); Q
+        (69*x^5 + 24*x^4 + 100*x^3 + 65*x^2 + 88*x + 97 : 65*x^5 + 28*x^4 + 5*x^3 + 45*x^2 + 42*x + 18 : 1)
+        sage: 8*Q == 0 and 4*Q != 0
+        True
+
     ::
 
         sage: from sage.schemes.elliptic_curves.ell_field import point_of_order
@@ -2078,10 +2085,23 @@
         (x : -Y : 1)
         sage: P.base_ring()
         Number Field in Y with defining polynomial Y^2 - x^3 - 7*x - 7 over its base field
+        sage: P.base_ring().base_field()
+        Number Field in x with defining polynomial x^4 + 14*x^2 + 28*x - 49/3
         sage: P.order()
         3
         sage: P.curve().a_invariants()
         (0, 0, 0, 7, 7)
+
+    ::
+
+        sage: Q = point_of_order(E, 4); Q  # random
+        (x : Y : 1)
+        sage: Q.base_ring()
+        Number Field in Y with defining polynomial Y^2 - x^3 - 7*x - 7 over its base field
+        sage: Q.base_ring().base_field()
+        Number Field in x with defining polynomial x^6 + 35*x^4 + 140*x^3 - 245*x^2 - 196*x - 735
+        sage: Q.order()
+        4
     """
     # Construct the field extension defined by the given polynomial,
     # in such a way that the result is recognized by Sage as a field.
@@ -2094,14 +2114,16 @@
             return poly.splitting_field(rng.variable_name())
         return fld.extension(poly, rng.variable_name())
 
-    l = ZZ(l)
-    if l == 1:
+    n = ZZ(n)
+    if n == 1:
         return E(0)
 
-    if not l.is_prime():
-        raise NotImplementedError('composite orders are currently unsupported')
+    l,m = n.is_prime_power(get_data=True)
+    if not m:
+        raise NotImplementedError('only prime-power orders are currently supported')
 
-    xpoly = E.division_polynomial(l)
+    xpoly = E.division_polynomial(n).radical()
+    xpoly //= E.division_polynomial(n//l).radical()
     if xpoly.degree() < 1:  # supersingular and l == p
         raise ValueError('curve does not have any points of the specified order')
 
@@ -2116,4 +2138,6 @@
         xx = FF(xx)
 
     EE = E.change_ring(FF)
-    return EE.lift_x(xx)
+    pt = EE.lift_x(xx)
+    pt.set_order(n, check=False)
+    return pt

src/sage/schemes/elliptic_curves/ell_point.py

@@ -1359,9 +1359,10 @@ def set_order(self, value=None, *, multiple=None, check=True):
                 raise ValueError('Value %s illegal for point order' % value)
             E = self.curve()
             q = E.base_ring().cardinality()
-            low, hi = Hasse_bounds(q)
-            if value > hi:
-                raise ValueError('Value %s illegal: outside max Hasse bound' % value)
+            if q < oo:
+                _, hi = Hasse_bounds(q)
+                if value > hi:
+                    raise ValueError('Value %s illegal: outside max Hasse bound' % value)
             if value * self != E(0):
                 raise ValueError('Value %s illegal: %s * %s is not the identity' % (value, value, self))
             if hasattr(self, '_order') and self._order != value:  # already known