add .torsion_basis() method to EllipticCurve_finite_field

src/sage/schemes/elliptic_curves/ell_field.py

@@ -910,6 +910,14 @@ def division_field(self, l, names='t', map=False, **kwds):
  sage: K.<v> = E.division_field(7); K
  Finite Field in v of size 433^16
 
+ .. SEEALSO::
+
+ To compute a basis of the `\ell`-torsion once the base field
+ has been extended, you may use
+ :meth:`sage.schemes.elliptic_curves.ell_number_field.EllipticCurve_number_field.torsion_subgroup`
+ or
+ :meth:`sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_finite_field.torsion_basis`.
+
  TESTS:
 
  Some random testing::

src/sage/schemes/elliptic_curves/ell_finite_field.py

@@ -1002,6 +1002,67 @@ def abelian_group(self):
  self.gens.set_cache(gens)
  return AdditiveAbelianGroupWrapper(self.point_homset(), gens, orders)
 
+ def torsion_basis(self, n):
+ r"""
+ Return a basis of the `n`-torsion subgroup of this elliptic curve,
+ assuming it is fully rational.
+
+ EXAMPLES::
+
+ sage: E = EllipticCurve(GF(62207^2), [1,0])
+ sage: E.abelian_group()
+ Additive abelian group isomorphic to Z/62208 + Z/62208 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 62207^2
+ sage: PA,QA = E.torsion_basis(2^8)
+ sage: PA.weil_pairing(QA, 2^8).multiplicative_order()
+ 256
+ sage: PB,QB = E.torsion_basis(3^5)
+ sage: PB.weil_pairing(QB, 3^5).multiplicative_order()
+ 243
+
+ ::
+
+ sage: E = EllipticCurve(GF(101), [4,4])
+ sage: E.torsion_basis(23)
+ Traceback (most recent call last):
+ ...
+ ValueError: curve does not have full rational 23-torsion
+ sage: F = E.division_field(23); F
+ Finite Field in t of size 101^11
+ sage: EE = E.change_ring(F)
+ sage: P, Q = EE.torsion_basis(23)
+ sage: P # random
+ (89*z11^10 + 51*z11^9 + 96*z11^8 + 8*z11^7 + 67*z11^6
+ + 31*z11^5 + 55*z11^4 + 59*z11^3 + 28*z11^2 + 8*z11 + 88
+ : 40*z11^10 + 33*z11^9 + 80*z11^8 + 87*z11^7 + 97*z11^6
+ + 69*z11^5 + 56*z11^4 + 17*z11^3 + 26*z11^2 + 69*z11 + 11
+ : 1)
+ sage: Q # random
+ (25*z11^10 + 61*z11^9 + 49*z11^8 + 17*z11^7 + 80*z11^6
+ + 20*z11^5 + 49*z11^4 + 52*z11^3 + 61*z11^2 + 27*z11 + 61
+ : 60*z11^10 + 91*z11^9 + 89*z11^8 + 7*z11^7 + 63*z11^6
+ + 55*z11^5 + 23*z11^4 + 17*z11^3 + 90*z11^2 + 91*z11 + 68
+ : 1)
+
+ .. SEEALSO::
+
+ Use :meth:`~sage.schemes.elliptic_curves.ell_field.EllipticCurve_field.division_field`
+ to determine a field extension containing the full `\ell`-torsion subgroup.
+
+ ALGORITHM:
+
+ This method currently uses :meth:`abelian_group` and
+ :meth:`AdditiveAbelianGroupWrapper.torsion_subgroup`.
+ """
+ # TODO: In many cases this is not the fastest algorithm.
+ # Alternatives include factoring division polynomials and
+ # random sampling (like PARI's ellgroup, but with a milder
+ # termination condition). We should implement these too
+ # and figure out when to use which.
+ T = self.abelian_group().torsion_subgroup(n)
+ if T.invariants() != (n, n):
+ raise ValueError(f'curve does not have full rational {n}-torsion')
+ return tuple(P.element() for P in T.gens())
+
  def is_isogenous(self, other, field=None, proof=True):
  """
  Return whether or not self is isogenous to other.

src/sage/schemes/elliptic_curves/ell_number_field.py

@@ -1961,6 +1961,11 @@ def torsion_subgroup(self):
  sage: EK = EllipticCurve([0,0,0,i,i+3])
  sage: EK.torsion_subgroup ()
  Torsion Subgroup isomorphic to Trivial group associated to the Elliptic Curve defined by y^2 = x^3 + i*x + (i+3) over Number Field in i with defining polynomial x^2 + 1 with i = 1*I
+
+ .. SEEALSO::
+
+ Use :meth:`~sage.schemes.elliptic_curves.ell_field.EllipticCurve_field.division_field`
+ to determine the field of definition of the `\ell`-torsion subgroup.
  """
  from .ell_torsion import EllipticCurveTorsionSubgroup
  return EllipticCurveTorsionSubgroup(self)