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

Let k be a field of characteristic zero containing a primitive fifth root of unity. Let 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 D₅ is a subgroup of Aut(X). We find that the intermediate Jacobian J(X) of X is isogenous to the product of an elliptic curve E and the self-product of an abelian surface B with real multiplication by Q(√5). We give explicit models of some algebraic curves related to the construction of J(X) as a Prym variety. This includes a two parameter family of curves of genus 2 whose Jacobians are isogenous to the abelian surfaces mentioned as above.