On characteristic polynomials of automorphisms of Enriques surfaces

Let $f$ be an automorphism of a complex Enriques surface $Y$ and let $p_f$ denote the characteristic polynomial of the isometry $f^*$ of the numerical N\'eron-Severi lattice of $Y$ induced by $f$. We apply a modification of McMullen's method to prove that the modulo-$2$ reduction $(p_f(x) \bmod 2)$ is a product of modulo-$2$ reductions of (some of) the five cyclotomic polynomials $\Phi_m$, where $m \leq 9$ and $m$ is odd. We study Enriques surfaces that realize modulo-$2$ reductions of $\Phi_7$, $\Phi_9$ and show that each of the five polynomials $(\Phi_m(x) \bmod 2)$ is a factor of the modulo-$2$ reduction $(p_f(x) \bmod 2)$ for a complex Enriques surface.