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

We prove that the moduli space X(1,7) of (1,7)-polarized abelian surfaces w ith canonical level-structure is birational to the Fano 3-fold V-22 of pola r 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 expl icitely the equations of the (1, 7)-very-ample-polarized abelian surfaces e mbedded in P-6.