New Families of ECM Curves for Cunningham Numbers

In this paper we study structures related to torsion of elliptic curves defined over number fields. The aim is to build families of elliptic curves more efficient to help factoring numbers of special form, including numbers from the Cunningham Project. We exhibit a family of curves with rational Z/4Z×Z/4Z torsion and positive rank over the field Q(ζ8) and a family of elliptic curves with rational Z/6Z× Z/3Z torsion and positive rank over the field Q(ζ3). These families have been used in finding new prime factors for the numbers 2 + 1 and 2 + 1. Along the way, we classify and give a parameterization of modular curves for some torsion subgroups.