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?

Notes on the Parity Conjectureby Tim Dokchitser inElliptic Curves, Hilbert Modular Forms and Galois Deformations, after remark 3.6. (an earlier version is available here, with a few typos). $\endgroup$ – Watson Oct 31 '18 at 19:09