Birational cubic fourfolds via an Enriques Cremona transformation

We describe a Cremona transformation on ${\mathbb{P}}^5$ given by the linear system of quintics that are double along the Fano model of an Enriques surface $S$. Using this Cremona transform, we construct a birational equivalence between the very general cubic fourfold $X$ of discriminant 44 and its unique Fourier–Mukai partner.