From: boyd@mahler.math.ubc.ca (David Boyd)
Newsgroups: sci.math.research
Subject: Re: Computing Weierstrass equations
Date: 26 Jun 1998 19:02:27 GMT

In article <6mvivq$m2$1@nnrp1.dejanews.com>,
Robin Chapman  <rjc@noether.ex.ac.uk> wrote:
>In article <6muf1h$8ej$1@nnrp1.dejanews.com>,
>  tchow@my-dejanews.com wrote:
>>
>> Is there freely available software that will take an elliptic curve over Q
>> that is given as the intersection of two quadrics (with explicitly given
>> equations) or as an equation of the form "y^2 = quartic" and will return a
>> Weierstrass equation for the curve?  I looked on Cremona's home page and it
>> seems that all his programs assume that the curve is already given in
>> Weierstrass form.
>
>There's a bit of a problem with this. In order to reduce a elliptic
>curve over Q to Weierstrass form, one first needs a rational point....
>
>I don't know of any software, but there are easy algorithms to reduce a curve
>in one of these forms, given a rational point, to Weierstrass form.
>See for instance Cassels, Lectures on Elliptic Curves (Cambridge, 1991)
>Chapter 8.
>

Suppose that  C: y^2 = a*x^4 + b*x^3 + c*x^2 + d*x + e,

Then the Jacobian of C has a Weierstrass equation

    E:    y^2 = x^3+c*x^2+(b*d-4*a*e)*x-(4*a*c*e-b^2*e-a*d^2); 

If C has a rational point, then it is birationally equivalent to E.

There are other forms but this has the advantage that the discriminant
of the cubic in E  is equal to the discriminant of the quartic in C.
  

David Boyd
Department of Mathematics
University of British Columbia
Vancouver, B.C., Canada
V6T 1Z2

boyd@math.ubc.ca