On the Mordell–Weil lattice of [Formula: see text] in characteristic 3

We study the elliptic curves given by [Formula: see text] 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 [Formula: see text] . In this way, using the Néron–Tate height on the Mordell–Weil group, we obtain lattices in dimension [Formula: see text] for every [Formula: see text] , which improve on the currently best known sphere packing densities in dimensions 162 (case [Formula: see text] ) and 486 (case [Formula: see text] ). For [Formula: see text] , the construction has the same packing density as the best currently known sphere packing in dimension 54, and for [Formula: see text] it has the same density as the lattice [Formula: see text] in dimension 6.