The systolic constant of orientable Bieberbach 3-manifolds

A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds are aspherical, therefore the supremum of their systolic ratio, over the set of Riemannian metrics, is finite by a fundamental result of M. Gromov. We study the optimal systolic ratio of compact of $3$-dimensional orientable Bieberbach manifolds which are not tori, and prove that it cannot be realized by a flat metric. We also highlight a metric that we construct on one type of such manifolds ($C_2$) which has interesting geometric properties : it is extremal in its conformal class and the systole is realized by "very many" geodesics.