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convenience method to construct elliptic-curve isomorphisms from u,r,s,t #37081

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convenience method to construct elliptic-curve isomorphisms from u,r,s,t #37081

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convenience method to construct elliptic-curve isomorphisms from u,r,s,t
yyyyx4 Jan 17, 2024
89f8531
mention default values for r,s,t
yyyyx4 Jan 18, 2024
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87 changes: 87 additions & 0 deletions src/sage/schemes/elliptic_curves/ell_generic.py
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Original file line number Diff line number Diff line change
Expand Up @@ -1555,6 +1555,93 @@ def scale_curve(self, u):
u = self.base_ring()(u) # because otherwise 1/u would round!
return self.change_weierstrass_model(1/u, 0, 0, 0)

def isomorphism(self, u, r=0, s=0, t=0, *, is_codomain=False):
r"""
Given four values `u,r,s,t` in the base ring of this curve, return
the :class:`WeierstrassIsomorphism` defined by `u,r,s,t` with this
curve as its codomain. (The value `u` must be a unit.)

Optionally, if the keyword argument ``is_codomain`` is set to ``True``,
return the isomorphism defined by `u,r,s,t` with this curve as its
**co**\domain.

EXAMPLES::

sage: E = EllipticCurve([1, 2, 3, 4, 5])
sage: iso = E.isomorphism(6); iso
Elliptic-curve morphism:
From: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
To: Elliptic Curve defined by y^2 + 1/6*x*y + 1/72*y = x^3 + 1/18*x^2 + 1/324*x + 5/46656 over Rational Field
Via: (u,r,s,t) = (6, 0, 0, 0)
sage: iso.domain() == E
True
sage: iso.codomain() == E.scale_curve(1 / 6)
True

sage: iso = E.isomorphism(1, 7, 8, 9); iso
Elliptic-curve morphism:
From: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
To: Elliptic Curve defined by y^2 + 17*x*y + 28*y = x^3 - 49*x^2 - 54*x + 303 over Rational Field
Via: (u,r,s,t) = (1, 7, 8, 9)
sage: iso.domain() == E
True
sage: iso.codomain() == E.rst_transform(7, 8, 9)
True

sage: iso = E.isomorphism(6, 7, 8, 9); iso
Elliptic-curve morphism:
From: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
To: Elliptic Curve defined by y^2 + 17/6*x*y + 7/54*y = x^3 - 49/36*x^2 - 1/24*x + 101/15552 over Rational Field
Via: (u,r,s,t) = (6, 7, 8, 9)
sage: iso.domain() == E
True
sage: iso.codomain() == E.rst_transform(7, 8, 9).scale_curve(1 / 6)
True

The ``is_codomain`` argument reverses the role of domain and codomain::

sage: E = EllipticCurve([1, 2, 3, 4, 5])
sage: iso = E.isomorphism(6, is_codomain=True); iso
Elliptic-curve morphism:
From: Elliptic Curve defined by y^2 + 6*x*y + 648*y = x^3 + 72*x^2 + 5184*x + 233280 over Rational Field
To: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
Via: (u,r,s,t) = (6, 0, 0, 0)
sage: iso.domain() == E.scale_curve(6)
True
sage: iso.codomain() == E
True

sage: iso = E.isomorphism(1, 7, 8, 9, is_codomain=True); iso
Elliptic-curve morphism:
From: Elliptic Curve defined by y^2 - 15*x*y + 90*y = x^3 - 75*x^2 + 796*x - 2289 over Rational Field
To: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
Via: (u,r,s,t) = (1, 7, 8, 9)
sage: iso.domain().rst_transform(7, 8, 9) == E
True
sage: iso.codomain() == E
True

sage: iso = E.isomorphism(6, 7, 8, 9, is_codomain=True); iso
Elliptic-curve morphism:
From: Elliptic Curve defined by y^2 - 10*x*y + 700*y = x^3 + 35*x^2 + 9641*x + 169486 over Rational Field
To: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
Via: (u,r,s,t) = (6, 7, 8, 9)
sage: iso.domain().rst_transform(7, 8, 9) == E.scale_curve(6)
True
sage: iso.codomain() == E
True

.. SEEALSO::

- :class:`~sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism`
- :meth:`rst_transform`
- :meth:`scale_curve`
"""
from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
if is_codomain:
return WeierstrassIsomorphism(None, (u,r,s,t), self)
return WeierstrassIsomorphism(self, (u,r,s,t))

# ###########################################################
#
# Explanation of the division (also known as torsion) polynomial
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