Moduli of (1,7)-polarized abelian surfaces via syzygies

We prove that the moduli space X(1,7) of (1,7)-polarized abelian surfaces with canonical level structure is birational to the Fano 3-fold V22 of polar hexagons of the Klein quartic X(7). In particular X(1,7) is rational and the birational map to P^3 is defined over Q. As a byproduct we obtain explicitely the equations of the (1,7)-very-ample polarized abelian surfaces in P^6.