Finding suitable curves for the elliptic curve method of factorization
A. O. L. Atkin, François Morain · Mathematics of Computation · 1993
Using the parametrizations of Kubert, we show how to produce infinite families of elliptic curves which have prescribed nontrivial torsion over Q and rank at least one. These curves can be used to speed up the ECM factorization algorithm of Lenstra. We also briefly discuss curves with complex multiplication in this context.