  We consider the optimization problem of maximizing the $k$-th Laplacian
eigenvalue, $\lambda_{k}$, over flat $d$-dimensional tori of fixed volume. For
$k=1$, this problem is equivalent to the densest lattice sphere packing
problem. For larger $k$, this is equivalent to the NP-hard problem of finding
the $d$-dimensional (dual) lattice with longest $k$-th shortest lattice vector.
As a result of extensive computations, for $d \leq 8$, we obtain a sequence of
flat tori, $T_{k,d}$, each of volume one, such that the $k$-th Laplacian
eigenvalue of $T_{k,d}$ is very large; for each (finite) $k$ the $k$-th
eigenvalue exceeds the value in (the $k\to \infty$ asymptotic) Weyl's law by a
factor between 1.54 and 2.01, depending on the dimension. Stationarity
conditions are derived and numerically verified for $T_{k,d}$ and we describe
the degeneration of the tori as $k \to \infty$.
