Integral Congruence Two Hyperbolic 5-Manifolds

In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form $H^5/\Gamma$ where $\Gamma$ is a torsion-free subgroup of minimal index of the congruence two subgroup $\Gamma^5_2$ of the group $\Gamma^5$ of positive units of the Lorentzian quadratic form $x_1^2+...+x_5^2-x_6^2$. We also show that $\Gamma^5_2$ is a reflection group with respect to a 5-dimensional right-angled convex polytope in $H^5$. As an application, we construct a hyperbolic 5-manifold of smallest known volume $7\zeta(3)/4$.