  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 $D_5$ is a subgroup of ${\rm Aut}(X)$. 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 $\mathbb{Q}(\sqrt{5})$. 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.
