Modular curves over number fields and ECM

Abstract

We construct families of elliptic curves defined over number fields and containing torsion groups $$\mathbb {Z}/{M_1}\mathbb {Z}\times \mathbb {Z}/{M_2}\mathbb {Z}$$ where $$(M_1, M_2)$$ 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 $$X_1(M_1, M_2)$$ 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 $$b^n-1$$ such that $$M_1 \mid n$$ . We explain how to find points of potentially large orders on the reduction, if we accept to use quadratic twists.