Does there exists a real number $C$ with the following property.
For any Enriques surface $E$ over a number field $K$ with K3 cover $X\to E$, there exists an ample divisor $H$ on $X$ such that $H^2 \leq C$?
Context: A polarization of degree $d$ on a K3 surface is an ample divisor $H$ such that $H^2 =d$. I'm wondering whether the polarization degree of a K3 surface with an Enriques involution is bounded.