  Starting from a beautiful idea of Kanev, we construct a uniformization of the
moduli space A_6 of principally polarized abelian 6-folds in terms of curves
and monodromy data. We show that the general ppav of dimension 6 is a
Prym-Tyurin variety corresponding to a degree 27 cover of the projective line
having monodromy the Weyl group of the E_6 lattice. Along the way, we establish
numerous facts concerning the geometry of the Hurwitz space of such E_6-covers,
including: (1) a proof that the canonical class of the Hurwitz space is big,
(2) a concrete geometric description of the Hodge-Hurwitz eigenbundles with
respect to the Kanev correspondence and (3) a description of the ramification
divisor of the Prym-Tyurin map from the Hurwitz space to A_6 in the terms of
syzygies of the Abel-Prym-Tyurin curve.
