We provide new explicit examples of lattice sphere packings in dimensions 54, 55, 162, 163, 486 and 487 that are the densest known so far, using Kummer families of elliptic curves over global function fields. In some cases, these families of elliptic curves have unbounded Mordell-Weil rank, and using the Néron-Tate height on the Mordell-Weil group, one can obtain lattices in high-dimensional euclidean spaces. In one case however, the rank of the curves happens to be bounded and non-zero in the Kummer family of (isotrivial) elliptic curves. In any case, these results rely on the explicit determination of the L-function of these curves, via Jacobi sums. This allows, under certain assumptions, to obtain formulas for the (analytic) rank of those curves.