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"
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".