  We provide a lower bound for the number of components of the algebraic
monodromy group in the situation of a $K3$ surface over a number field $k$. In
the CM case, our bound is sharp. As an application, we describe, in the case of
CM, the jump character \cite[Definition~2.4.6]{CEJ} entirely in terms of the
endomorphism field and the geometric Picard rank.
