Computing the rational torsion of an elliptic curve using tate normal form

It is a classical result (apparently due to Tate) that all elliptic curves with a torsion point of order n (4 less than or equal to n less than or equal to 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. (C) 2002 Elsevier Science (USA).