Compute composite degree (separable) isogenies of EllipticCurves

src/sage/schemes/elliptic_curves/ell_field.py

@@ -1598,9 +1598,9 @@ def kernel_polynomial_from_divisor(self, f, l, *, check=True):
 
     def isogenies_prime_degree(self, l=None, max_l=31):
         """
-        Return a list of all separable isogenies of given prime degree(s)
-        with domain equal to ``self``, which are defined over the base
-        field of ``self``.
+        Return a list of all separable isogenies (up to post-composition with
+        isomorphisms) of given prime degree(s) with domain equal to ``self``,
+        which are defined over the base field of ``self``.
 
         INPUT:
 
@@ -1882,6 +1882,113 @@ def isogenies_prime_degree(self, l=None, max_l=31):
         from .isogeny_small_degree import isogenies_prime_degree
         return sum([isogenies_prime_degree(self, d) for d in L], [])
 
+    def isogenies_degree(self, n, *, _intermediate=False):
+        r"""
+        Return an iterator of all separable isogenies of given degree (up to
+        post-composition with isomorphisms) with domain equal to ``self``,
+        which are defined over the base field of ``self``.
+
+        ALGORITHM:
+
+        The prime factors `p` of `n` are processed one by one, each time
+        "branching" out by taking isogenies of degree `p`.
+
+        INPUT:
+
+        - ``n`` -- integer, or its
+          :class:`~sage.structure.factorization.Factorization`
+
+        - ``_intermediate`` -- (bool, default: False): If set, the curves
+          traversed within the depth-first search are returned. This is for
+          internal use only.
+
+        EXAMPLES::
+
+            sage: E = EllipticCurve(GF(11), [1, 1])
+            sage: list(E.isogenies_degree(23 * 19))
+            [Composite morphism of degree 437 = 19*23:
+               From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 11
+               To:   Elliptic Curve defined by y^2 = x^3 + 8*x + 7 over Finite Field of size 11,
+             Composite morphism of degree 437 = 19*23:
+               From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 11
+               To:   Elliptic Curve defined by y^2 = x^3 + 2*x + 6 over Finite Field of size 11,
+             Composite morphism of degree 437 = 19*23:
+               From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 11
+               To:   Elliptic Curve defined by y^2 = x^3 + 6*x + 2 over Finite Field of size 11,
+             Composite morphism of degree 437 = 19*23:
+               From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 11
+               To:   Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 11]
+
+        ::
+
+            sage: E = EllipticCurve(GF(next_prime(2^32)), j=1728)
+            sage: sorted([phi.codomain().j_invariant() for phi in E.isogenies_degree(11 * 17 * 19^2)])
+            [1348157279, 1348157279, 1713365879, 1713365879, 3153894341, 3153894341, 3153894341,
+            3153894341, 3225140514, 3225140514, 3673460198, 3673460198, 3994312564, 3994312564,
+            3994312564, 3994312564]
+            sage: it = E.isogenies_degree(2^2); it
+            <generator object EllipticCurve_field.isogenies_degree at 0x...>
+            sage: all(phi.degree() == 2^2 for phi in it)
+            True
+
+        ::
+
+            sage: pol = PolynomialRing(QQ, 'x')([1, -3, 5, -5, 5, -3, 1])
+            sage: L.<a> = NumberField(pol)
+            sage: js = hilbert_class_polynomial(-23).roots(L, multiplicities=False)
+            sage: len(js)
+            3
+            sage: E = EllipticCurve(j=js[0])
+            sage: len(list(E.isogenies_degree(2**2)))
+            7
+            sage: len(list(E.isogenies_degree(2**5))) # long time (15s)
+            99
+
+        TESTS::
+
+            sage: E = EllipticCurve(GF(next_prime(2^32)), j=1728)
+            sage: list(E.isogenies_degree(2^2, _intermediate=True))
+            [Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 4294967311
+               Via:  (u,r,s,t) = (1, 0, 0, 0),
+             Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 4294967311 to Elliptic Curve defined by
+              y^2 = x^3 + 4294967307*x over Finite Field of size 4294967311,
+             Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 4294967311 to Elliptic Curve defined by
+              y^2 = x^3 + 4294967307*x over Finite Field of size 4294967311,
+             Composite morphism of degree 4 = 2^2:
+               From: Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 4294967311
+               To:   Elliptic Curve defined by y^2 = x^3 + 16*x over Finite Field of size 4294967311,
+             Composite morphism of degree 4 = 2^2:
+               From: Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 4294967311
+               To:   Elliptic Curve defined by y^2 = x^3 + 4294967267*x + 4294967199 over Finite Field of size 4294967311,
+             Composite morphism of degree 4 = 2^2:
+               From: Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 4294967311
+               To:   Elliptic Curve defined by y^2 = x^3 + 4294967267*x + 112 over Finite Field of size 4294967311]
+
+        The following curve has no degree-`5` isogenies, so the code is quick::
+
+            sage: E = EllipticCurve(GF(103), [3, 5])
+            sage: list(E.isogenies_degree(5 * product(prime_range(7, 100))))
+            []
+        """
+        from sage.structure.factorization import Factorization
+        from sage.schemes.elliptic_curves.weierstrass_morphism import identity_morphism
+
+        if not isinstance(n, Factorization):
+            n = Integer(n).factor()
+
+        if n.value() == 1:
+            yield identity_morphism(self)
+            return
+
+        p = n[-1][0]
+        for iso in self.isogenies_degree(n / p, _intermediate=_intermediate):
+            if _intermediate:
+                yield iso
+
+            Eiso = iso.codomain()
+            for next_iso in Eiso.isogenies_prime_degree(p):
+                yield next_iso * iso
+
     def is_isogenous(self, other, field=None):
         """
         Return whether or not self is isogenous to other.