K3 surfaces, cyclotomic polynomials and orthogonal groups

Let X be a complex projective K3 surface and let T-X be its transcendental lattice; the characteristic polynomials of isometries of T-X induced by automorphisms of X are powers of cyclotomic polynomials. Which powers of cyclotomic polynomials occur? The aim of this note is to answer this question, as well as related ones, and give an alternative approach to some results of Kondo, Machida, Oguiso, Vorontsov, Xiao and Zhang; this leads to questions and results concerning orthogonal groups of lattices.