From: jr@redmink.demon.co.uk (John R Ramsden)
Subject: Re: Converting to Weierstrass normal form
Date: Mon, 20 Sep 1999 15:18:49 GMT
Newsgroups: sci.math
Keywords: example of a transformation to an elliptic curve to Weierstrass form

On Sun, 19 Sep 1999 16:19:47 -0700, "Randall L. Rathbun" <randallr@abac.com>
wrote:

>Can someone show how to change the following equation to the Weierstrass
>normal form?
>k is a constant, x,y are the variables.
>
>(1)    xy = k(x+1)(x-1)(y+1)(y-1)
>
>We should have some final result like:
>
>     y^2 + axy + by = x^3 + cx^2 + dx + e
>
>where a,b,c,d,e are the Weierstrass constants.
>
>Thanks!

Define z by:

   y  =  (z + x)/(z - x).

Replacing this in (1) gives:

   y  =  4.k.(x^2 - 1).z / (z - x)^2

and equating the two expressions for y gives:

   4.k.(x^2 - 1).z  =  z^2 - x^2

or equivalently:

   (1 + 4.k.z).x^2  =  z.(z + 4.k)

Then defining:

   t  =  x.(1 + 4.k.z)

you have your Weirstrauss normal form as:

   t^2  =  z.(z + 4.k).(4.k.z + 1)

and it is fully reducible as well. That may be handy for you!


Cheers

---

John R Ramsden             #   "No one who has not shared a submarine
                           #    with a camel can claim to have plumbed
 (jr@redmink.demon.co.uk)  #    the depths of human misery."
                           #
                           #      Ritter von Haske
                           #        "Adventures of a U-boat Commander".