diophan wrote:
To get back to the question, did you find the inverse by hand or is
there something in Sage to help out? I have potentially a large number
of cubics I'd like to carry this out with and if there's a way to avoid
doing it by hand each time that'd be great.
Ahem, ever heard of tab completion?
sage: R.<x,y,z> = QQ[]
sage: f = 3*y^2*x-y^2*z-2*x*y*z+y*z^2+2*x^3-2*x^2*z
sage: e = EllipticCurve_from_cubic(f,[0,0,1])
sage: e
Scheme morphism:
From: Closed subscheme of Projective Space of dimension 2 over
Rational Field defined by:
2*x^3 + 3*x*y^2 - 2*x^2*z - 2*x*y*z - y^2*z + y*z^2
To: Elliptic Curve defined by y^2 + 6*x*y + 256*y = x^3 - 73*x^2
over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(1/8*x*y - 1/16*y^2 - 1/8*y*z : -x^2 + 1/8*x*y + 3/16*y^2 + x*z
+ 3/8*y*z : -1/256*y^2)
sage: type(e)
<class
'sage.schemes.elliptic_curves.weierstrass_transform.WeierstrassTransformationWithInverse_class'>
sage: e.inverse()
Scheme morphism:
From: Elliptic Curve defined by y^2 + 6*x*y + 256*y = x^3 - 73*x^2
over Rational Field
To: Closed subscheme of Projective Space of dimension 2 over
Rational Field defined by:
2*x^3 + 3*x*y^2 - 2*x^2*z - 2*x*y*z - y^2*z + y*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(-12*x*z - 4*y*z : 32*x*z : x^2 - 28*x*z - 4*y*z)
sage: e.inverse().defining_polynomials()
[-12*x*z - 4*y*z, 32*x*z, x^2 - 28*x*z - 4*y*z]
-leif
On Saturday, May 24, 2014 4:38:48 PM UTC-4, Nils Bruin wrote:
On Saturday, May 24, 2014 9:18:29 AM UTC-7, Volker Braun wrote:
Its a 4:1 map so you can't invert it...
I would find that surprising. For a general plane cubic, there are
good recipes for getting a 9:1 map to a Weierstrass model in general
and a 1:1 map when a rational point is specified. A 4:1 map is
rather unnatural to get in that situation. You'd expect that from a
y^2=quartic in x model.
Indeed, the map returned is invertible, the inverse being:
[ -12*x*z - 4*y*z, 32*x*z, x^2 - 28*x*z - 4*y*z]
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