Automorphisms of certain Niemeier lattices and Elliptic Fibrations

Nishiyama introduced a lattice theoretic classification of the elliptic fibrations on a $K3$ surface. In a previous paper we used his method to exhibit $52$ elliptic fibrations, up to isomorphisms, of the singular $K3$ surface of discriminant $-12$. We prove here that the list is complete with a $53$th fibration, thanks to a remark of Elkies and Sch\"{u}tt. We characterize the fibration both theoretically and with a Weierstrass model.