Computing the Rational Torsion of an Elliptic Curve Using Tate Normal Form

Abstract

It is a classical result (apparently due to Tate) that all elliptic curves with a torsion point of order n(4⩽ n⩽10, or n=12) lie in a one-parameter family. However, this fact does not appear to have been used ever for computing the torsion of an elliptic curve. We present here an extremely down-to-earth algorithm using the existence of such a family.