On the Mordell-Weil lattice of \(y^2=x^3+bx+t^{3^n+1}\) in characteristic 3 (Q2122583)

The author considers the elliptic curve defined by the Weierstrass equation \[ y^2=x^3+bx+t^{3^n+1} \] over the global function field \(\mathbb{F}_{3^n}(t)\). For certain values of the parameter \(b\), an explicit computation of the \(L\)-function of the latter curve is provided. As an application, the arithmetic of these elliptic curves is employed to obtain better sphere packing densities than the current ones.