#3416: Weierstrass form and Jacobian for cubics and certain other genus-one
curves
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Reporter: moretti |
Owner: was
Type: enhancement |
Status: needs_review
Priority: major |
Milestone: sage-5.7
Component: elliptic curves |
Resolution:
Keywords: nagell, weierstrass, cubic, elliptic curves, editor_wstein |
Work issues:
Report Upstream: N/A |
Reviewers: John Cremona, Marco Streng, Nils Bruin
Authors: Niels Duif, Volker Braun |
Merged in:
Dependencies: |
Stopgaps:
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Changes (by {'newvalue': u'Niels Duif, Volker Braun', 'oldvalue': u'Niels
Duif'}):
* status: needs_work => needs_review
* author: Niels Duif => Niels Duif, Volker Braun
Old description:
> Implement transformations to put a general cubic (with a point) into
> Weierstrass form.
>
> Apply:
>
> * [attachment:trac_3416_elliptic_curve_from_cubic3.patch]
>
> See [attachment:cubic_to_weierstrass_documentation.pdf] for details on
> how the algorithm works.
New description:
Implement transformations to put a general cubic (with a point) into
Weierstrass form:
{{{
sage: R.<u,v,w> = QQ[]
sage: EllipticCurve(u^3+v^3+w^3, [1,-1,0])
Elliptic Curve defined by y^2 + 2*x*y - 1/3*y = x^3 - x^2 + 1/3*x - 1/27
over Rational Field
sage: EllipticCurve_from_cubic(u^3+v^3+w^3, [1,-1,0])
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)
}}}
Also, Jacobians (without specifying point):
{{{
sage: R.<u,v,w> = QQ[]
sage: Jacobian(u^3+v^3+w^3)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: Jacobian(u^3+v^3+w^3, morphism=True)
Scheme morphism:
From: Projective Curve over Rational Field defined by u^3 + v^3 + w^3
To: Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
Defn: Defined on coordinates by sending (u : v : w) to
(u*v^7*w + u*v^4*w^4 + u*v*w^7 : v^9 + 3/2*v^6*w^3 - 3/2*v^3*w^6 -
w^9 : -v^6*w^3 - v^3*w^6)
}}}
Apply:
* [attachment:trac_3416_elliptic_curve_from_cubic_vb.patch]
* [attachment:trac_3416_jacobians.patch]
See [attachment:cubic_to_weierstrass_documentation.pdf] for details on how
the algorithm works.
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/3416#comment:51>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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