Coverage for local/lib/python2.7/site-packages/sage/schemes/elliptic_curves/weierstrass_transform.py : 100%
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""" Morphism to bring a genus-one curve into Weierstrass form
You should use :func:`~sage.schemes.elliptic_curves.constructor.EllipticCurve_from_cubic` or :func:`~sage.schemes.elliptic_curves.constructor.EllipticCurve_from_curve` to construct the transformation starting with a cubic or with a genus one curve.
EXAMPLES::
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) """
############################################################################## # Copyright (C) 2013 Volker Braun <vbraun.name@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ ##############################################################################
r""" A morphism of a 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: 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) """
""" Return the homogeneous rescaling to apply after the coordinate substitution.
OUTPUT:
A polynomial. See the example below.
EXAMPLES::
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
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: 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 """
defining_polynomials, post_multiplication, inv_defining_polynomials, inv_post_multiplication): """ Construct morphism of a a genus-one curve to/from the Weierstrass form with its inverse.
EXAMPLES::
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) """ domain, codomain, defining_polynomials, post_multiplication) codomain, domain, inv_defining_polynomials, inv_post_multiplication)
""" Return the inverse.
OUTPUT:
A morphism in the opposite direction. This may be a rational inverse or an analytic inverse.
EXAMPLES::
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) """
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