On the Mordell-Weil lattice of in characteristic 3

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