  Given a very general abelian fivefold $A$ and a principal polarization
$\Theta \subset A$, we construct surfaces generating the algebraic part of the
middle cohomology $H^4(\Theta, {\mathbb Q})$, and determine the intersection
pairing between these surfaces. In particular, we obtain a new proof of the
Hodge conjecture for $H^4(\Theta, {\mathbb Q})$ and show that it contains a
copy of the root lattice of $E_6$.
