In the work of A. P. Ogg, Elliptic Curves and Wild Ramification, he proves that the conductor of an elliptic curve is independent of the choice of $\ell$. That is, for example, if $E$ is an elliptic curve defined over $Q$ and if $G_{Q_p}$ is the Galois group of the local field $Q_p$. When $p\neq \ell$, then the conductor of the $\ell$-adic $G_{Q_p}$-representation induced by the $\ell$-adic Tate module of $E$ is independent of the choice of $\ell$.
My question is: is the statement "the conductor of the $\ell$-adic $G_{Q_p}$-representation $\rho_{\ell}$ is independent of the choice of $\ell$" always true as long as we know that $\rho_{\ell}$ is coming from an etale cohomology of a smooth algerabic variety? If not, how much do we know about it? Any comments and suggestions are welcome.