Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization

TL;DR

This paper introduces methods for constructing elliptic curves with large torsion groups and positive rank over small degree number fields, highlighting their potential use in the elliptic curve factorization method (ECM).

Contribution

The paper presents new techniques for constructing elliptic curves with large torsion and positive rank over small degree number fields, expanding the tools available for ECM.

Findings

Methods for constructing elliptic curves with large torsion over small degree fields

Identification of elliptic curves with positive rank suitable for ECM

Potential applications of these curves in factorization algorithms

Abstract

In this paper, we present several methods for construction of elliptic curves with large torsion group and positive rank over number fields of small degree. We also discuss potential applications of such curves in the elliptic curve factorization method (ECM).