The Points of a Certain Fivefold over Finite Fields and the Twelfth Power of the Eta Function

If p is an odd prime, then denote by p the field with p elements. We prove that a certain fivefold is modular in the sense that for every odd p, the number of its points over p is predicted explicitly by the pth coefficient of the Fourier expansion of the weight 6 modular form η12(2z).