Cubic fourfolds with a symplectic automorphism of prime order

We classify the algebraic and transcendental lattices of a general cubic fourfold with a symplectic automorphism of prime order. We prove the rationality of two families of cubic fourfolds, and that they are not equivariantly rational with respect to their group of automorphisms. As an application, we classify the cohomological action of symplectic birational transformations of manifolds of OG10 type that are induced by prime order sympletic automorphisms of cubic fourfolds.