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".