Cubic Surfaces with Special Periods

We show that the vector of period ratios of a cubic surface is rational over $Q(ω)$, where $ω= \exp(2πi/3)$ if and only if the associate abelian variety is isogeneous to a product of Fermat elliptic curves. We also show how to construct cubic surfaces from a suitable totally real quintic number field $K_0$. The ring of rational endomorphisms of the associated abelian variety is $K = K_0(ω)$.

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