On the Mordell-Weil lattice of y 2 = x 3 + b x + t 3 n + 1 in characteristic 3.

We study the elliptic curves given by y 2 = x 3 + b x + t 3 n + 1 over global function fields of characteristic 3 ; in particular we perform an explicit computation of the L -function by relating it to the zeta function of a certain superelliptic curve u 3 + b u = v 3 n + 1 . In this way, using the Néron-Tate height on the Mordell-Weil group, we obtain lattices in dimension 2 · 3 n for every n ≥ 1 , which improve on the currently best known sphere packing densities in dimensions 162 (case n = 4 ) and 486 (case n = 5 ). For n = 3 , the construction has the same packing density as the best currently known sphere packing in dimension 54, and for n = 1 it has the same density as the lattice E 6 in dimension 6.