Mordell-Weil lattices in characteristic 2, III: A Mordell-Weil lattice of rank 128

In the first paper of this series, we constructed a family of lattices in dimensions 2^{n+1} for positive integers n, and proved that the associated lattice packings of spheres equal or exceed the previous records for several values of n. In this paper we analyze the case n=6, which is now the first case where the resulting lattice MW128 yields a previously unknown lattice of record density. We prove that the elliptic curve used to construct this lattice has trivial Tate-Shafarevich group, and exhibit on that curve a rational point whose canonical height attains our lower bound on the minimal norm of MW128. We thus obtain the normalized center density of the lattice. We then report on a computation that determined all the rational points of minimal norm, thus obtaining the kissing number of MW128. Like the packing density, the kissing number of MW128 is by a considerable factor the largest known kissing number of a lattice in this dimension.