Let $E$ be an elliptic curve over $\mathbb{Q}_\ell$ and consider the image of the $\ell$-adic representation. Is there a description of this image similar to Serre's description of the image of the mod $\ell$ representation? (I.e., which depends on whether $E$ has e.g. bad multiplicative/good ordinary/good supersingular reduction?)

I'm looking for a description similar to Serre's description of the mod $\ell$ image (see e.g. Lemma 3.4 of this).