Let $R$ be a discrete valuation ring with fraction field $K$ (valuation being $v$) and finite residue field $k$, $l$ be a rational prime invertible in $R$, $D(\bar{v})$ be a decomposition subgroup of the absolute Galois group $G=Gal(K_s/K)$, $\sigma\in D(\bar{v})$ be an element whose image in $Gal(\bar{k}/k)$ is the Frobenius.
Let $A/K$ be an abelian variety with potential good reduction. Consider the Galois representation $\rho_l:G\to GL(T_lA)$. Then Theorem 3, p.499 of the classical paper Good reduction of abelian varieties by Serre-Tate says the characteristic polynomial of $\rho_l(\sigma)$ has coefficients in $\mathbb{Z}$ independent of $l$. Its roots are Weil numbers. For me this conclusion is very strong, so I am wondering how $\rho_l(\sigma)$ depends on the choice of $\sigma$. If we choose another $\sigma'$ over Frobenius, do we know $\rho_l(\sigma),\rho_l(\sigma')$ share same characteristic polynomial?