On the PSL2(F19)-invariant cubic sevenfold

It has been proved by Adler that there exists a unique cubic hypersurface X7 in P8 which is invariant under the action of the simple group PSL2(F19). In the present note we study the intermediate Jacobian of X7 and in particular we prove that the subjacent 85-dimensional torus is an Abelian variety. The symmetry group G=PSL2(F19) defines uniquely a G-invariant Abelian 9-fold A(X7), which we study in detail and describe its period lattice.