Let $G=Gal(\bar K/K)$ be the absolute Galois group of a number field $K$. Let $v$ be a finite place of $K$ and $w$ a place of $\bar K$ extending $v$. Take an $\ell$-adic representation $$ \rho: G \longrightarrow \mathrm{GL}_n(\mathbb{Q}_{\ell}).$$
Assume $\rho$ is unramified at $v$ and denote with $F_{w,\rho}$ the image of the Frobenius element associated at $w$.
How the conjugacy class of $F_{w,\rho}$ does depend only by $v$?