Classification of torsion of elliptic curves over quintic fields

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: $\mathbb{Z}/28\mathbb{Z}$, $\mathbb{Z}/30\mathbb{Z}$