Closed orientable three manifold with finite cover by $S^1 \times S^2$ or $T^3$

Question

I have been thinking about a problem where I can conclude that I have a closed orientable three manifold which is covered by $S^1 \times S^2$ or $S^1 \times T^2$. I think that the geometrization theory allows me to describe the possible manifolds explicitly, but not being so confident with that theory I am wondering if someone could tell me what are the possibilities as well as some explanation.