  We compute characteristic numbers of elliptically fibered fourfolds with
multisections or non-trivial Mordell-Weil groups. We first consider the models
of type E$_{9-d}$ with $d=1,2,3,4$ whose generic fibers are normal elliptic
curves of degree $d$. We then analyze the characteristic numbers of the
$Q_7$-model, which provides a smooth model for elliptic fibrations of rank one
and generalizes the E$_5$, E$_6$, and E$_7$-models. Finally, we examine the
characteristic numbers of $G$-models with $G=\text{SO}(n)$ with $n=3,4,5,6$ and
$G=\text{PSU}(3)$ whose Mordell-Weil groups are respectively
$\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3 \mathbb{Z}$. In each case, we
compute the Chern and Pontryagin numbers, the Euler characteristic, the
holomorphic genera, the Todd-genus, the L-genus, the A-genus, and the
eight-form curvature invariant from M-theory.
