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We study the elliptic curves given by $y^2 = x^3 + b x + t^{3^n+1}$ overglobal function fields of characteristic $3$; in particular we perform anexplicit computation of the $L$-function by relating it to the zeta function ofa certain superelliptic curve $u^3 + b u = v^{3^n + 1}$.  In this way, using the N\'eron-Tate height on the Mordell--Weil group, weobtain lattices in dimension $2 \cdot 3^n$ for every $n \geq 1$, which improveon 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 packingdensity 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 $6$.

On the Mordell--Weil lattice of $y^2 = x^3 + b x + t^{3^n + 1}$ in characteristic $3$