A geometric approach to elliptic curves with torsion groups $\mathbb{Z}/10\mathbb{Z}$, $\mathbb{Z}/12\mathbb{Z}$, $\mathbb{Z}/14\mathbb{Z}$, and $\mathbb{Z}/16\mathbb{Z}$

TL;DR

This paper introduces new geometric parametrisations of elliptic curves with specific torsion groups over rationals and quadratic fields, leading to the discovery of infinite families and record ranks for certain torsion groups.

Contribution

It provides new geometric parametrisations for elliptic curves with torsion groups Z/10Z, Z/12Z, Z/14Z, and Z/16Z, and finds record ranks for some of these groups.

Findings

Discovered three infinite families of Z/12Z torsion curves with positive rank

Achieved a rank 3 elliptic curve with Z/14Z torsion, a new record

Found new elliptic curves with Z/16Z torsion and rank 3

Abstract

We give new parametrisations of elliptic curves in Weierstrass normal form $y^2=x^3+ax^2+bx$ with torsion groups $\mathbb{Z}/10\mathbb{Z}$ and $\mathbb{Z}/12\mathbb{Z}$ over $\mathbb{Q}$, and with $\mathbb{Z}/14\mathbb{Z}$ and $\mathbb{Z}/16\mathbb{Z}$ over quadratic fields. Even though the parametrisations are equivalent to those given by Kubert and Rabarison, respectively, with the new parametrisations we found three infinite families of elliptic curves with torsion group $\mathbb{Z}/12\mathbb{Z}$ and positive rank. Furthermore, we found elliptic curves with torsion group $\mathbb{Z}/14\mathbb{Z}$ and rank $3$, which is a new record for such curves, as well as some new elliptic curves with torsion group $\mathbb{Z}/16\mathbb{Z}$ and rank $3$.