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 $\mathbb{Z}/16\mathbb{Z}$

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 $3$.