Classification of torsion of elliptic curves over sextic fields

Abstract

We determine all the groups that appear as the torsion group of an elliptic curve over a number field of degree $6$. Apart from the $44$ groups that occur infinitely often, which were determined by Derickx and Sutherland, exactly four groups occur: $\mathbb{Z}/25\mathbb{Z}$, $\mathbb{Z}/37\mathbb{Z}$, $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}$. The first three were known to occur, while the group $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}$ is new. We also show that, up to isomorphism, each of the groups $\mathbb{Z}/37\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}$ is the torsion group of a single elliptic curve over a single sextic field. The methods build on those developed by Derickx and Najman for quartic fields and by the author for quintic fields. Among the new ingredients are a formal immersion criterion at the prime $2$ which uses the Cartier operator, and arguments with several diamond operators and with cuspidal divisor classes on quotients of modular curves.