From: elkies@ramanujan.math.harvard.edu (Noam Elkies)
Newsgroups: sci.math.research
Subject: Re: Torsion group of  y^2= x( x^2 + ax + b )
Date: 8 Feb 1995 16:28:53 GMT

In article <3h9h4c$kcd@jhunix1.hcf.jhu.edu>
kwon@chow.mat.jhu.edu (Soonhak Kwon) writes:
>Are there any well known theorem about  Torsion Group of elliptic
>curve of the following form, 
>       y^2 = x(x^2+ax+b)  where a,b are integers
>Please e mail me if any of you know about this.
>I'd be very apreciated.

A celebrated theorem of Mazur(*) states that the only possible
torsion groups for elliptic curves over Q are the cyclic groups
of order 1,2,3,4,5,6,7,8,9,10,12 [sic -- 11 is not possible]
and (Z/2)*(Z/2n) for n=1,2,3,4.  The curves y^2 = x(x^2+ax+b)
are simply those constrained to have a 2-torsion point at (0,0),
so of the above torsion groups all arise except the five of
odd order 1,3,5,7,9.

Examples of all the possible torsion groups may be found in
Cremona's tables (see his _Algorithms for Modular Elliptic Curves_,
Cambridge Univ. Press 1992); for instance the largest group  Z/2 * Z/8
occurs for the curve 210-E2:  y^2 + xy = x^3 - 1070 x + 7812,
a.k.a.  Y^2 = X (X-64) (4X-175).

(*) Mazur, B.: Rational isogenies of prime degree, 
nventiones Math. 44 (1978), 129--162.

--Noam D. Elkies (elkies@zariski.harvard.edu)
  Dept. of Mathematics, Harvard University