Modular curves over number fields and ECM

We construct families of elliptic curves defined over number fields and containing torsion groups Z/ M₁Z× Z/ M₂Z where (M₁, M₂) belongs to { (1 , 11) , (1, 14), (1, 15), (2, 10), (2, 12), (3, 9), (4, 8), (6 , 6) } (i.e., when the corresponding modular curve X₁(M₁, M₂) has genus 1). We provide formulae for the curves and give examples of number fields for which the corresponding elliptic curves have non-zero ranks, giving explicit generators using D. Simon’s program whenever possible. 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 bⁿ- 1 such that M₁∣ n. We explain how to find points of potentially large orders on the reduction, if we accept to use quadratic twists.