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Let $k$ be a field of characteristic zero containing a primitive fifth rootof unity. Let $X/k$ be a smooth cubic threefold with an automorphism of orderfive, then we observe that over a finite extension of the field actually thedihedral group $D_5$ is a subgroup of ${\rm Aut}(X)$. We find that theintermediate Jacobian $J(X)$ of $X$ is isogenous to the product of an ellipticcurve $E$ and the self-product of an abelian surface $B$ with realmultiplication by $\mathbb{Q}(\sqrt{5})$. We give explicit models of somealgebraic curves related to the construction of $J(X)$ as a Prym variety. Thisincludes a two parameter family of curves of genus 2 whose Jacobians areisogenous to the abelian surfaces mentioned as above.

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