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Journal Articles Research in Number Theory Year : 2022

Modular curves over number fields and ECM

Abstract

We construct families of elliptic curves defined over number fields and containing torsion groups Z=M1Z x Z=M2Z where (M1;M2) belongs to f(1; 11), (1; 14), (1; 15), (2; 10), (2; 12), (3; 9), (4; 8), (6; 6)g (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 j n. We explain how to find points of potentially large orders on the reduction, if we accept to use quadratic twists.
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Dates and versions

hal-03606355 , version 1 (11-03-2022)
hal-03606355 , version 2 (09-01-2023)

Identifiers

  • HAL Id : hal-03606355 , version 2

Cite

François Morain. Modular curves over number fields and ECM. Research in Number Theory, 2022. ⟨hal-03606355v2⟩
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