Mathematical Research Letters

Volume 23 (2016)

Number 1

Torsion of rational elliptic curves over cubic fields and sporadic points on $X_1(n)$

Pages: 245 – 272



Filip Najman (Department of Mathematics, University of Zagreb, Croatia)


We classify the possible torsion structures of rational elliptic curves over cubic fields. Along the way we find a previously unknown torsion structure over a cubic field, $\mathbb{Z} / 21 \mathbb{Z}$, which corresponds to a sporadic point on $X_1(21)$ of degree 3, which is the lowest possible degree of a sporadic point on a modular curve $X_1(n)$.


elliptic curves, torsion subgroups, cubic fields, modular curves

2010 Mathematics Subject Classification

11G05, 11G18, 11G25

Accepted 1 September 2014

Published 25 May 2016