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

Flat tori with large Laplacian eigenvalues in dimensions up to eight