  We study a K3 surface, which appears in the two-loop mixed
electroweak-quantum chromodynamic virtual corrections to Drell--Yan scattering.
A detailed analysis of the geometric Picard lattice is presented, computing its
rank and discriminant in two independent ways: first using explicit divisors on
the surface and then using an explicit elliptic fibration. We also study in
detail the elliptic fibrations of the surface and use them to provide an
explicit Shioda--Inose structure. Moreover, we point out the physical relevance
of our results.
