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# 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

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n1.a12

#### Author

#### Abstract

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 $X_1(n)$.

#### Keywords

elliptic curves, torsion subgroups, cubic fields, modular curves

#### 2010 Mathematics Subject Classification

11G05, 11G18, 11G25