  The systole of a compact non simply connected Riemannian manifold is the
smallest length of a non-contractible closed curve ; the systolic ratio is the
quotient $(\mathrm{systole})^n/\mathrm{volume}$. Its supremum on the set of all
the riemannian metrics, is known to be finite for a large class of manifolds,
including the $K(\pi,1)$. We study the optimal systolic ratio of compact,
3-dimensional non orientable Bieberbach manifolds, and prove that it cannot be
realized by a flat metric.
