the interesting part of the cohomology of the theta divisor $d$ of an abelian fivefold $a$ shares numerical properties with the lie algebra $e_6$. we define 27 surfaces inside $d$, one for each realisation of $a$ as a prym variety, and explain how they generate a sublattice of $h^4(d, \mathbb{z})$ isomorphic to the root lattice of $e_6$. this gives an effective proof of the hodge conjecture for the theta divisor.