Morphism to bring a genus-one curve into Weierstrass form¶

You should use EllipticCurve_from_cubic() or EllipticCurve_from_curve() to construct the transformation starting with a cubic or with a genus one curve.

EXAMPLES:

Sage

sage: R.<u,v,w> = QQ[]
sage: f = EllipticCurve_from_cubic(u^3 + v^3 + w^3, [1,-1,0], morphism=True);  f
Scheme morphism:
  From: Projective Plane Curve over Rational Field defined by u^3 + v^3 + w^3
  To:   Elliptic Curve defined by y^2 - 9*y = x^3 - 27 over Rational Field
  Defn: Defined on coordinates by sending (u : v : w) to
        (-w : 3*u : 1/3*u + 1/3*v)

sage: finv = f.inverse();  finv
Scheme morphism:
  From: Elliptic Curve defined by y^2 - 9*y = x^3 - 27 over Rational Field
  To:   Projective Plane Curve over Rational Field defined by u^3 + v^3 + w^3
  Defn: Defined on coordinates by sending (x : y : z) to
        (1/3*y : -1/3*y + 3*z : -x)

sage: (u^3 + v^3 + w^3)(f.inverse().defining_polynomials()) * f.inverse().post_rescaling()
-x^3 + y^2*z - 9*y*z^2 + 27*z^3

sage: E = finv.domain()
sage: E.defining_polynomial()(f.defining_polynomials()) * f.post_rescaling()
u^3 + v^3 + w^3

sage: f([1,-1,0])
(0 : 1 : 0)
sage: f([1,0,-1])
(3 : 9 : 1)
sage: f([0,1,-1])
(3 : 0 : 1)

Python

>>> from sage.all import *
>>> R = QQ['u, v, w']; (u, v, w,) = R._first_ngens(3)
>>> f = EllipticCurve_from_cubic(u**Integer(3) + v**Integer(3) + w**Integer(3), [Integer(1),-Integer(1),Integer(0)], morphism=True);  f
Scheme morphism:
  From: Projective Plane Curve over Rational Field defined by u^3 + v^3 + w^3
  To:   Elliptic Curve defined by y^2 - 9*y = x^3 - 27 over Rational Field
  Defn: Defined on coordinates by sending (u : v : w) to
        (-w : 3*u : 1/3*u + 1/3*v)

>>> finv = f.inverse();  finv
Scheme morphism:
  From: Elliptic Curve defined by y^2 - 9*y = x^3 - 27 over Rational Field
  To:   Projective Plane Curve over Rational Field defined by u^3 + v^3 + w^3
  Defn: Defined on coordinates by sending (x : y : z) to
        (1/3*y : -1/3*y + 3*z : -x)

>>> (u**Integer(3) + v**Integer(3) + w**Integer(3))(f.inverse().defining_polynomials()) * f.inverse().post_rescaling()
-x^3 + y^2*z - 9*y*z^2 + 27*z^3

>>> E = finv.domain()
>>> E.defining_polynomial()(f.defining_polynomials()) * f.post_rescaling()
u^3 + v^3 + w^3

>>> f([Integer(1),-Integer(1),Integer(0)])
(0 : 1 : 0)
>>> f([Integer(1),Integer(0),-Integer(1)])
(3 : 9 : 1)
>>> f([Integer(0),Integer(1),-Integer(1)])
(3 : 0 : 1)

Sage Live

R.<u,v,w> = QQ[]
f = EllipticCurve_from_cubic(u^3 + v^3 + w^3, [1,-1,0], morphism=True);  f
finv = f.inverse();  finv
(u^3 + v^3 + w^3)(f.inverse().defining_polynomials()) * f.inverse().post_rescaling()
E = finv.domain()
E.defining_polynomial()(f.defining_polynomials()) * f.post_rescaling()
f([1,-1,0])
f([1,0,-1])
f([0,1,-1])

class sage.schemes.elliptic_curves.weierstrass_transform.WeierstrassTransformation(domain, codomain, defining_polynomials, post_multiplication)[source]¶

Bases: SchemeMorphism_polynomial

A morphism of a genus-one curve to/from the Weierstrass form.

INPUT:

domain, codomain – two schemes, one of which is an elliptic curve
defining_polynomials – triplet of polynomials that define the transformation
post_multiplication – a polynomial to homogeneously rescale after substituting the defining polynomials

EXAMPLES:

Sage

sage: P2.<u,v,w> = ProjectiveSpace(2,QQ)
sage: C = P2.subscheme(u^3 + v^3 + w^3)
sage: E = EllipticCurve([2, -1, -1/3, 1/3, -1/27])
sage: from sage.schemes.elliptic_curves.weierstrass_transform import WeierstrassTransformation
sage: f = WeierstrassTransformation(C, E, [w, -v-w, -3*u-3*v], 1);  f
Scheme morphism:
  From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
  u^3 + v^3 + w^3
  To:   Elliptic Curve defined by y^2 + 2*x*y - 1/3*y = x^3 - x^2 + 1/3*x - 1/27
        over Rational Field
  Defn: Defined on coordinates by sending (u : v : w) to
        (w : -v - w : -3*u - 3*v)

sage: f([-1, 1, 0])
(0 : 1 : 0)
sage: f([-1, 0, 1])
(1/3 : -1/3 : 1)
sage: f([ 0,-1, 1])
(1/3 : 0 : 1)

sage: A2.<a,b> = AffineSpace(2,QQ)
sage: C = A2.subscheme(a^3 + b^3 + 1)
sage: f = WeierstrassTransformation(C, E, [1, -b-1, -3*a-3*b], 1);  f
Scheme morphism:
  From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  a^3 + b^3 + 1
  To:   Elliptic Curve defined by y^2 + 2*x*y - 1/3*y
        = x^3 - x^2 + 1/3*x - 1/27 over Rational Field
  Defn: Defined on coordinates by sending (a, b) to
        (1 : -b - 1 : -3*a - 3*b)
sage: f([-1,0])
(1/3 : -1/3 : 1)
sage: f([0,-1])
(1/3 : 0 : 1)

Python

>>> from sage.all import *
>>> P2 = ProjectiveSpace(Integer(2),QQ, names=('u', 'v', 'w',)); (u, v, w,) = P2._first_ngens(3)
>>> C = P2.subscheme(u**Integer(3) + v**Integer(3) + w**Integer(3))
>>> E = EllipticCurve([Integer(2), -Integer(1), -Integer(1)/Integer(3), Integer(1)/Integer(3), -Integer(1)/Integer(27)])
>>> from sage.schemes.elliptic_curves.weierstrass_transform import WeierstrassTransformation
>>> f = WeierstrassTransformation(C, E, [w, -v-w, -Integer(3)*u-Integer(3)*v], Integer(1));  f
Scheme morphism:
  From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
  u^3 + v^3 + w^3
  To:   Elliptic Curve defined by y^2 + 2*x*y - 1/3*y = x^3 - x^2 + 1/3*x - 1/27
        over Rational Field
  Defn: Defined on coordinates by sending (u : v : w) to
        (w : -v - w : -3*u - 3*v)

>>> f([-Integer(1), Integer(1), Integer(0)])
(0 : 1 : 0)
>>> f([-Integer(1), Integer(0), Integer(1)])
(1/3 : -1/3 : 1)
>>> f([ Integer(0),-Integer(1), Integer(1)])
(1/3 : 0 : 1)

>>> A2 = AffineSpace(Integer(2),QQ, names=('a', 'b',)); (a, b,) = A2._first_ngens(2)
>>> C = A2.subscheme(a**Integer(3) + b**Integer(3) + Integer(1))
>>> f = WeierstrassTransformation(C, E, [Integer(1), -b-Integer(1), -Integer(3)*a-Integer(3)*b], Integer(1));  f
Scheme morphism:
  From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  a^3 + b^3 + 1
  To:   Elliptic Curve defined by y^2 + 2*x*y - 1/3*y
        = x^3 - x^2 + 1/3*x - 1/27 over Rational Field
  Defn: Defined on coordinates by sending (a, b) to
        (1 : -b - 1 : -3*a - 3*b)
>>> f([-Integer(1),Integer(0)])
(1/3 : -1/3 : 1)
>>> f([Integer(0),-Integer(1)])
(1/3 : 0 : 1)

Sage Live

P2.<u,v,w> = ProjectiveSpace(2,QQ)
C = P2.subscheme(u^3 + v^3 + w^3)
E = EllipticCurve([2, -1, -1/3, 1/3, -1/27])
from sage.schemes.elliptic_curves.weierstrass_transform import WeierstrassTransformation
f = WeierstrassTransformation(C, E, [w, -v-w, -3*u-3*v], 1);  f
f([-1, 1, 0])
f([-1, 0, 1])
f([ 0,-1, 1])
A2.<a,b> = AffineSpace(2,QQ)
C = A2.subscheme(a^3 + b^3 + 1)
f = WeierstrassTransformation(C, E, [1, -b-1, -3*a-3*b], 1);  f
f([-1,0])
f([0,-1])

post_rescaling()[source]¶

Return the homogeneous rescaling to apply after the coordinate substitution.

OUTPUT: a polynomial; see the example below

EXAMPLES:

Sage

sage: R.<a,b,c> = QQ[]
sage: cubic =  a^3+7*b^3+64*c^3
sage: P = [2,2,-1]
sage: f = EllipticCurve_from_cubic(cubic, P, morphism=True).inverse()
sage: f.post_rescaling()
-1/7

Python

>>> from sage.all import *
>>> R = QQ['a, b, c']; (a, b, c,) = R._first_ngens(3)
>>> cubic =  a**Integer(3)+Integer(7)*b**Integer(3)+Integer(64)*c**Integer(3)
>>> P = [Integer(2),Integer(2),-Integer(1)]
>>> f = EllipticCurve_from_cubic(cubic, P, morphism=True).inverse()
>>> f.post_rescaling()
-1/7

Sage Live

R.<a,b,c> = QQ[]
cubic =  a^3+7*b^3+64*c^3
P = [2,2,-1]
f = EllipticCurve_from_cubic(cubic, P, morphism=True).inverse()
f.post_rescaling()

So here is what it does. If we just plug in the coordinate transformation, we get the defining polynomial up to scale. This method returns the overall rescaling of the equation to bring the result into the standard form:

Sage

sage: cubic(f.defining_polynomials())
7*x^3 - 7*y^2*z + 1806336*y*z^2 - 155373797376*z^3
sage: cubic(f.defining_polynomials()) * f.post_rescaling()
-x^3 + y^2*z - 258048*y*z^2 + 22196256768*z^3

Python

>>> from sage.all import *
>>> cubic(f.defining_polynomials())
7*x^3 - 7*y^2*z + 1806336*y*z^2 - 155373797376*z^3
>>> cubic(f.defining_polynomials()) * f.post_rescaling()
-x^3 + y^2*z - 258048*y*z^2 + 22196256768*z^3

Sage Live

cubic(f.defining_polynomials())
cubic(f.defining_polynomials()) * f.post_rescaling()

sage.schemes.elliptic_curves.weierstrass_transform.WeierstrassTransformationWithInverse(domain, codomain, defining_polynomials, post_multiplication, inv_defining_polynomials, inv_post_multiplication)[source]¶

Construct morphism of a genus-one curve to/from the Weierstrass form with its inverse.

EXAMPLES:

Sage

sage: R.<u,v,w> = QQ[]
sage: f = EllipticCurve_from_cubic(u^3 + v^3 + w^3, [1,-1,0], morphism=True);  f
Scheme morphism:
  From: Projective Plane Curve over Rational Field defined by u^3 + v^3 + w^3
  To:   Elliptic Curve defined by y^2 - 9*y = x^3 - 27 over Rational Field
  Defn: Defined on coordinates by sending (u : v : w) to
    (-w : 3*u : 1/3*u + 1/3*v)

Scheme morphism:
  From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
  u^3 + v^3 + w^3
  To:   Elliptic Curve defined by y^2 + 2*x*y + 1/3*y
        = x^3 - x^2 - 1/3*x - 1/27 over Rational Field
  Defn: Defined on coordinates by sending (u : v : w) to
        (-w : -v + w : 3*u + 3*v)

Python

>>> from sage.all import *
>>> R = QQ['u, v, w']; (u, v, w,) = R._first_ngens(3)
>>> f = EllipticCurve_from_cubic(u**Integer(3) + v**Integer(3) + w**Integer(3), [Integer(1),-Integer(1),Integer(0)], morphism=True);  f
Scheme morphism:
  From: Projective Plane Curve over Rational Field defined by u^3 + v^3 + w^3
  To:   Elliptic Curve defined by y^2 - 9*y = x^3 - 27 over Rational Field
  Defn: Defined on coordinates by sending (u : v : w) to
    (-w : 3*u : 1/3*u + 1/3*v)

Scheme morphism:
  From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
  u^3 + v^3 + w^3
  To:   Elliptic Curve defined by y^2 + 2*x*y + 1/3*y
        = x^3 - x^2 - 1/3*x - 1/27 over Rational Field
  Defn: Defined on coordinates by sending (u : v : w) to
        (-w : -v + w : 3*u + 3*v)

Sage Live

R.<u,v,w> = QQ[]
f = EllipticCurve_from_cubic(u^3 + v^3 + w^3, [1,-1,0], morphism=True);  f

class sage.schemes.elliptic_curves.weierstrass_transform.WeierstrassTransformationWithInverse_class(domain, codomain, defining_polynomials, post_multiplication)[source]¶

Bases: WeierstrassTransformation

inverse()[source]¶

Return the inverse.

OUTPUT:

A morphism in the opposite direction. This may be a rational inverse or an analytic inverse.

EXAMPLES:

Sage

sage: R.<u,v,w> = QQ[]
sage: f = EllipticCurve_from_cubic(u^3 + v^3 + w^3, [1,-1,0], morphism=True)
sage: f.inverse()
Scheme morphism:
  From: Elliptic Curve defined by y^2 - 9*y = x^3 - 27 over Rational Field
  To:   Projective Plane Curve over Rational Field defined by u^3 + v^3 + w^3
  Defn: Defined on coordinates by sending (x : y : z) to
  (1/3*y : -1/3*y + 3*z : -x)

Python

>>> from sage.all import *
>>> R = QQ['u, v, w']; (u, v, w,) = R._first_ngens(3)
>>> f = EllipticCurve_from_cubic(u**Integer(3) + v**Integer(3) + w**Integer(3), [Integer(1),-Integer(1),Integer(0)], morphism=True)
>>> f.inverse()
Scheme morphism:
  From: Elliptic Curve defined by y^2 - 9*y = x^3 - 27 over Rational Field
  To:   Projective Plane Curve over Rational Field defined by u^3 + v^3 + w^3
  Defn: Defined on coordinates by sending (x : y : z) to
  (1/3*y : -1/3*y + 3*z : -x)

Sage Live

R.<u,v,w> = QQ[]
f = EllipticCurve_from_cubic(u^3 + v^3 + w^3, [1,-1,0], morphism=True)
f.inverse()