The characteristic polynomial of Frobenius of an RM curve

Question

Let $C$ be a genus two curve over $\mathbb{Q}$. We can reduce $C$ modulo a prime $p$ to obtain a curve $\bar{C}$ over $\mathbb{F}_p$. By counting points of $\bar{C}$ over $\mathbb{F}_p$ and $\mathbb{F}_{p^2}$, we can construct the characteristic polynomial of Frobenius $f_p$. If $C$ has RM by a number field $K$, then, apparently, the polynomial $f_p$ splits over $K$. I've seen this mentioned a number of times, but I've never seen any referencing. Does anyone know a good reference for this?