Modular curves over number fields and ECM

Abstract

We use genus 1 modular curves X1(M1, M2) where M1 | M2 to construct families of elliptic curves defined over number fields and containing torsion groups Z/M1Z × Z/M2Z. The reductions of these curves can be used to speed up ECM for factoring numbers with special properties, a typical example being (factors of) Cunningham numbers b n − 1 such that n | M1.