On intermediate Jacobians of cubic threefolds admitting an automorphism of order five

Let k k be a field of characteristic zero containing a primitive fifth root of unity. Let X/k X/k be a smooth cubic threefold with an automorphism of order five, then we observe that over a finite extension of the field actually the dihedral group D 5 D5 is a subgroup of Aut(X) Aut(X) . We find that the intermediate Jacobian J(X) J(X) of X X is isogenous to the product of an elliptic curve E E and the self-product of an abelian surface B B with real multiplication by Q(5 – √ ) Q(5) . We give explicit models of some algebraic curves related to the construction of J(X) J(X) as a Prym variety. This includes a two parameter family of curves of genus 2 2 whose Jacobians are isogenous to the abelian surfaces mentioned as above.