Rational curves on the supersingular K3 surface with Artin invariant 1 in characteristic 3

We show the existence of 112 non-singular rational curves on the supersingular K3 surface X with Artin invariant 1 in characteristic 3 by several ways. These non-singular rational curves have the minimum degree with respect to a very ample divisor on X. Using these rational curves, we have a (16)10-configuration and a (2804,11210)-configuration on the K3 surface. Moreover we study the Picard lattice by using the theory of the Leech lattice. The 112 non-singular rational curves correspond to 112 Leech roots.