ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 02 May 2017 12:21:31 -0500transform quartic to weierstrasshttp://ask.sagemath.org/question/37478/transform-quartic-to-weierstrass/ If we have a quartic surface like $y^2=a(t)*x^4+b(t)*x^3+...+e(t)$ with a point on it and want to transform it to weierstrass surface (cubic surface) with sage , what can we do? is it possible?Mon, 01 May 2017 03:33:26 -0500http://ask.sagemath.org/question/37478/transform-quartic-to-weierstrass/Answer by FrédéricC for <p>If we have a quartic surface like $y^2=a(t)<em>x^4+b(t)</em>x^3+...+e(t)$ with a point on it and want to transform it to weierstrass surface (cubic surface) with sage , what can we do? is it possible?</p>
http://ask.sagemath.org/question/37478/transform-quartic-to-weierstrass/?answer=37487#post-id-37487Like that:
sage: t = polygen(QQ, 't')
sage: x, y = polygens(FractionField(t.parent()), 'x,y')
sage: E = Curve(-y**2+x**4+x**3+t*x**2+(1+t)*x+(4+3*t))
sage: Jacobian(E)
Elliptic Curve defined by y^2 = x^3 + (-1/3*t^2-11*t-15)*x + (2/27*t^3-22/3*t^2-6*t+5) over Fraction Field of Univariate Polynomial Ring in t over Rational FieldTue, 02 May 2017 12:21:31 -0500http://ask.sagemath.org/question/37478/transform-quartic-to-weierstrass/?answer=37487#post-id-37487Answer by dan_fulea for <p>If we have a quartic surface like $y^2=a(t)<em>x^4+b(t)</em>x^3+...+e(t)$ with a point on it and want to transform it to weierstrass surface (cubic surface) with sage , what can we do? is it possible?</p>
http://ask.sagemath.org/question/37478/transform-quartic-to-weierstrass/?answer=37485#post-id-37485After a translation in the variable $x$ we can assume that the point on the quartic is $(0,q)$. Then plugging in this point into the equation, we get an equation of the shape:
$$ y^2 = Ax^4+Bx^3+Cx^2+Dx+q^2\ .$$
For the simplicity of the following formula, let us write $Q=2q$.
The following transformation is given in [Ian Conell, Elliptic Curves Hanbook, page 105, Quartic to Weierstrass] . (Wonderful exposition.)
$$ x = ( Q(v+q)+Du )/u^2\ ,$$
$$ y = ( Q^2(v+q) + Q(Cu^2+Du) - D^2u^2/Q)/u^3\ .$$
(For a sage example with "simpler" coeficients: [https://ask.sagemath.org/question/36637/change-of-variable-from-hyperellictic-curve-to-weierstrass-form/](https://ask.sagemath.org/question/36637/change-of-variable-from-hyperellictic-curve-to-weierstrass-form/)
(This is of course a community wiki post... All karma goes to all beautiful books and people in this field.)Tue, 02 May 2017 02:34:13 -0500http://ask.sagemath.org/question/37478/transform-quartic-to-weierstrass/?answer=37485#post-id-37485