Let $E$ be an elliptic curve defined over $\mathbf{Q}$, fix an odd prime $p>3$, let $T\_p$ denote the $p$-adic Tate module of $E$, and let $V\_p = T\_p \otimes \mathbf{Q}\_p$.


If the action of $G\_\mathbf{Q}$ is unramified at $\ell$, it is known that the characteristic polynomial of Frobenius, $\mathrm{Frob}\_\ell$, is


$x^2 - a\_\ell(E)x + \ell$.


Now let $\ell$ be a ramified prime for this representation, and let $(V\_p)\_{I\_\ell}$ be the maximal quotient of $V\_p$ on which the $\ell$-inertia group $I\_\ell$ acts trivially (so it is one-dimensional). Then it makes sense to ask the following question.



> 
> Is it possible to describe the eigenvalue of Frob$\_\ell$ on $(V\_p)\_{I\_\ell}$ in terms of explicit data from the elliptic curve, as in the unramified case?
> 
> 
> 



Frobenius at ramified primes
Are you looking for the following sort of answer? $a\_\ell(E)=1$ if $E$ has split multiplicative reduction, $a\_\ell(E)=-1$ if $E$ has non-split multiplicative reduction, and $a\_\ell(E)=0$ if $E$ has additive reduction. This gives the "right" local factors for the $L$-series, i.e., $L\_p=1\pm p^{-s}$ for multiplicative reduction and $L\_p=1$ for additive reduction.


