A parametrised family of Mordell curves

TL;DR

This paper introduces a parametrized family of Mordell curves with a minimum rank of three and a torsion group of Z/3Z, expanding understanding of their algebraic properties.

Contribution

It presents a new parametrized family of Mordell curves with guaranteed rank and specific torsion structure, advancing the classification of these elliptic curves.

Findings

Family of Mordell curves with rank ≥ 3

Torsion group is Z/3Z for these curves

Provides explicit parametrizations of such curves

Abstract

An elliptic curve defined by an equation of the type $y^2=x^3+d$ is called a Mordell curve. We obtain a parametrised family of Mordell curves whose rank, in general, is at least three, and whose torsion group is $\mathbb{Z}/3\mathbb{Z}$.