Classification of torsion of elliptic curves over quintic fields

RESUMO

We determine all the groups that appear as the torsion group of an elliptic curve over a quintic number field. Apart from the groups that already occur infinitely often, which were determined by Derickx and Sutherland, exactly three groups occur: $\mathbb{Z}/28\mathbb{Z}$, $\mathbb{Z}/30\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$. Up to isomorphism of the pair $(K,E)$, the first and the third are each realised by a single elliptic curve and the second by two curves, which are $2$-isogenous over a common quintic field. The curves realising $\mathbb{Z}/28\mathbb{Z}$ and $\mathbb{Z}/30\mathbb{Z}$ were found by van Hoeij, while the group $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$ is new, and $5$ is the smallest degree in which a non-cyclic sporadic torsion group occurs. The methods improve on those developed by Derickx and Najman, and are based on Hecke sieves and the arithmetic of cuspidal divisor classes.