The Fermat curve of fourth degree (Q1208090)

Let \(F_ 4:X^ 4+Y^ 4+Z^ 4=0\) be the Fermat curve of degree 4 over \(\mathbb{Q}\). Each automorphism of the curve \(F_ 4\) induces an endomorphism of the corresponding jacobian variety \(J_ 4\). This defines a map \(\mathbb{Q} [\text{Aut} (F_ 4)] \to \text{End} (J_ 4) \bigotimes \mathbb{Q}\). The author shows by explicit calculations that \(\text{End} (J_ 4)\) is precisely the image of the group ring \(\mathbb{Z} [\text{Aut} (J_ 4)]\). It is then a consequence that \(J_ 4\) is not isomorphic to a product of elliptic curves.