## Fields of definition of elliptic curves with prescribed torsion

### Volume 181 / 2017

Acta Arithmetica 181 (2017), 85-95
MSC: Primary 11G05; Secondary 11G18.
DOI: 10.4064/aa170323-20-9
Published online: 27 October 2017

#### Abstract

We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined over $\mathbb{Q}$. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.