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Enable creation of minimal Weierstrass models from cubics #38744

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Enable creation of minimal Weierstrass models from cubics #38744

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b55c1c3
Update elliptic_curves/constructor.py
jcbland Sep 26, 2024
6408d09
Merge branch 'develop' into minimal-cubic
kwankyu Sep 30, 2024
7fda240
Update constructor.py
jcbland Sep 30, 2024
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Remove extra whitespace
jcbland Oct 2, 2024
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Merge branch 'develop' into minimal-cubic
jcbland Nov 7, 2024
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118 changes: 115 additions & 3 deletions src/sage/schemes/elliptic_curves/constructor.py
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Original file line number Diff line number Diff line change
Expand Up @@ -791,7 +791,7 @@ def coefficients_from_j(j, minimal_twist=True):
return Sequence([0, 0, 0, -3*j*k, -2*j*k**2], universe=K)


def EllipticCurve_from_cubic(F, P=None, morphism=True):
def EllipticCurve_from_cubic(F, P=None, morphism=True, minimize=False):
r"""
Construct an elliptic curve from a ternary cubic with a rational point.

Expand All @@ -816,6 +816,10 @@ def EllipticCurve_from_cubic(F, P=None, morphism=True):
returns a birational isomorphism from `C` to a Weierstrass
elliptic curve `E`, otherwise just returns `E`

- ``minimize`` -- boolean (default: ``False``). If ``True``
the elliptic curve `E` will be a minimal model.


OUTPUT:

Either (when ``morphism``=``False``) an elliptic curve `E` in long
Expand Down Expand Up @@ -919,6 +923,39 @@ def EllipticCurve_from_cubic(F, P=None, morphism=True):
sage: E.defining_polynomial()(f.defining_polynomials()) * f.post_rescaling()
a^3 + b^3 + 60*c^3

We can get the birational isomorphism to and from the minimal model
of the elliptic curve::

sage: fmin = EllipticCurve_from_cubic(cubic, P, morphism=True, minimize=True)
sage: fmin
Scheme morphism:
From: Projective Plane Curve over Rational Field defined by a^3 + b^3 + 60*c^3
To: Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field
Defn: Defined on coordinates by sending (a : b : c) to
(-c : 3/2*a - 3/2*b : 1/180*a + 1/180*b)

sage: fmininv = fmin.inverse(); fmininv
Scheme morphism:
From: Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field
To: Projective Plane Curve over Rational Field defined by a^3 + b^3 + 60*c^3
Defn: Defined on coordinates by sending (x : y : z) to
(1/3*y + 90*z : -1/3*y + 90*z : -x)

We check that `fmin` maps `P` to the origin::

sage: fmin([1,-1,0])
(0 : 1 : 0)
sage: fmininv([0,1,0])
(-1 : 1 : 0)

We check that `fmin` maps the cubic to the minimal model::

sage: cubic(fmininv.defining_polynomials()) * fmininv.post_rescaling()
-x^3 + y^2*z + 24300*z^3
sage: Emin = fmin.codomain()
sage: Emin.defining_polynomial()(fmin.defining_polynomials()) * fmin.post_rescaling()
a^3 + b^3 + 60*c^3

If the given point is not a flex and the cubic has no rational
flexes, then the cubic can not be transformed to a Weierstrass
equation by a linear transformation. The general birational
Expand Down Expand Up @@ -986,6 +1023,46 @@ def EllipticCurve_from_cubic(F, P=None, morphism=True):
sage: [finv(Q) for Q in E.torsion_points() if Q]
[(9 : -9/4 : 1), (-9 : 0 : 1), (0 : 1 : 0)]

We can also map to the minimal model of this curve::

sage: E.minimal_model()
Elliptic Curve defined by y^2 = x^3 - 147*x - 286 over Rational Field
sage: fmin = EllipticCurve_from_cubic(cubic, [1,-1,1], morphism=True, minimize=True); fmin
Scheme morphism:
From: Projective Plane Curve over Rational Field defined
by x^2*y + 4*x*y^2 + x^2*z + 8*x*y*z + 4*y^2*z + 9*x*z^2 + 9*y*z^2
To: Elliptic Curve defined
by y^2 = x^3 - 147*x - 286 over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(2352637/19906560000*x^2 - 2352637/31104000000*x*y
- 30584281/622080000000*y^2 + 44700103/82944000000*x*z
+ 16468459/51840000000*y*z - 101163391/276480000000*z^2
: 2352637/5529600000*x^2 + 44700103/13824000000*x*y
+ 2352637/576000000*x*z
+ 16468459/4608000000*y*z + 2352637/1024000000*z^2
: -2352637/99532800000*x^2 - 2352637/124416000000*x*y
- 2352637/622080000000*y^2 + 2352637/82944000000*x*z
+ 2352637/207360000000*y*z - 2352637/276480000000*z^2)
sage: fmin([-4, 1, 0])
(-7 : 20 : 1)
sage: [fmin(Q) for Q in [[9, -9/4, 1], [-9, 0, 1], [0, 1, 0]]]
[(-11 : 0 : 1), (-2 : 0 : 1), (13 : 0 : 1)]
sage: fmininv = fmin.inverse(); fmininv
Scheme morphism:
From: Elliptic Curve defined
by y^2 = x^3 - 147*x - 286 over Rational Field
To: Projective Plane Curve over Rational Field defined
by x^2*y + 4*x*y^2 + x^2*z + 8*x*y*z + 4*y^2*z + 9*x*z^2 + 9*y*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(33177600000000/312900721*x^2 - 1559347200000000/312900721*x*z
+ 33177600000000/312900721*y*z + 14664499200000000/312900721*z^2
: -33177600000000/312900721*x^2 + 564019200000000/312900721*x*z
- 132710400000000/312900721*y*z + 66355200000000/16468459*z^2
: 33177600000000/312900721*x^2 - 33177600000000/44700103*x*z
- 33177600000000/312900721*y*z - 2587852800000000/312900721*z^2)
sage: fmininv([-7, 20, 1])
(-4 : 1 : 0)

In this example, the given point ``P`` is not a flex but the cubic
does have a rational flex, ``(-4:0:1)``. We return a linear
isomorphism which maps this flex to the point at infinity on the
Expand Down Expand Up @@ -1192,7 +1269,13 @@ def EllipticCurve_from_cubic(F, P=None, morphism=True):
F3 = F2([-x, y/b, z*a*b]) / a
# assert F3.coefficient(x**3) == -1
# assert F3.coefficient(y*y*z) == 1
E = EllipticCurve(F3([x,y,1]))
E0 = EllipticCurve(F3([x,y,1]))

if minimize:
E = E0.minimal_model()
else:
E = E0

if not morphism:
return E

Expand Down Expand Up @@ -1248,7 +1331,13 @@ def EllipticCurve_from_cubic(F, P=None, morphism=True):
S2 = a*S1(T2)
# assert F4 == F(xyzW) // S2

E = EllipticCurve(F4([x,y,1]))
E0 = EllipticCurve(F4([x,y,1]))

if minimize:
E = E0.minimal_model()
else:
E = E0

if not morphism:
return E

Expand All @@ -1266,6 +1355,29 @@ def EllipticCurve_from_cubic(F, P=None, morphism=True):
# assert F4(fwd_defining_poly)*fwd_post == F

# Construct the morphism
if minimize:
fwd_defining_poly0 = fwd_defining_poly
fwd_post0 = fwd_post
inv_defining_poly0 = inv_defining_poly
inv_post0 = inv_post

phi = E0.isomorphism_to(E)
T3 = [x*phi.u**2 + phi.r*z,
y*phi.u**3 + phi.s*x*phi.u**2 + phi.t*z,
z]
T3I = [(x - phi.r*z) / (phi.u**2),
(y - phi.s*(x - phi.r*z) - phi.t*z) / (phi.u**3),
z] # inverse of T3
inv_defining_poly = [t(T3) for t in inv_defining_poly0]
fwd_defining_poly = [t(fwd_defining_poly0) for t in T3I]

if flex_point is not None:
inv_post = inv_post0
fwd_post = fwd_post0
else:
inv_post = inv_post0(T3)
xyzImin = [t(xyzI) for t in T3I]
fwd_post = a/(x*z*z)(xyzImin)

return WeierstrassTransformationWithInverse(
C, E, fwd_defining_poly, fwd_post, inv_defining_poly, inv_post)
Expand Down
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