Elliptic curves with torsion group $\Z /6\Z $

TL;DR

This paper constructs and analyzes families of elliptic curves with torsion group Z/6Z, demonstrating they have generic rank at least 3 and identifying specific examples with record rank 8 over the rationals.

Contribution

It provides new explicit families of elliptic curves with torsion Z/6Z and rank at least 3, including the current record rank 8 over Q, and determines their Mordell-Weil groups.

Findings

Families with torsion Z/6Z and rank ≥3 constructed.

Exact rank over Q(t) is 3 for these families.

Record rank 8 curves over Q identified.

Abstract

We exhibit several families of elliptic curves with torsion group isomorphic to $ \Z/6\Z$ and generic rank at least $3$. Families of this kind have been constructed previously by several authors: Lecacheux, Kihara, Eroshkin and Woo. We mention the details of some of them and we add other examples developed more recently by Dujella and Peral, and MacLeod. Then we apply an algorithm of Gusi\'c and Tadi\'c and we find the exact rank over $\Q(t)$ to be 3 and we also determine free generators of the Mordell-Weil group for each family. By suitable specializations, we obtain the known and new examples of curves over $\Q$ with torsion $ \Z/6\Z$ and rank $8$, which is the current record.