Let $K$ be an extension of $\mathbb{Q}_p$ and $E$ be an elliptic curve over $K$ with potentially good reduction, i.e. there exists a finite extension $L/K$ such that $E/L$ has good reduction.
Let $\rho: G_K \to \operatorname{GL}_2(\mathbb{C})$ be the Galois representation of $E$.
Question Which values of $\rho$ determine the whole representation?
As an example, we look at Example 5 of Tim and Vladimir Dokchitser's Euler Factors determine local Weil Representation:
The authors claim that the elliptic curve $$E/ K=\mathbb{Q}_{13}: \quad y^2 = x^3-26x $$ has good reduction over $L=\mathbb{Q}_{13}(\sqrt[4]{13})$ (which I understand) and, as $L/K$ is a totally ramified extension of degree $4$, the representation factors through $$ \operatorname{Gal}(L^{nr}/K) \simeq \hat{\mathbb{Z}} \times C_4$$ (which I have not understood yet). Let $g$ be a generator of $C_4$. Since any Frobenius element $\operatorname{Frob}_L$ on $L$ generates $\hat{\mathbb{Z}}$, we only need to know $\rho(\operatorname{Frob}_L)$ and $\rho(g)$ to determine the whole representation $\rho$.
Additional Question: Is there a way to generalize this observation?
Any help is appreciated!
You mean $\ell \ne p$ prime and $\rho:G_K\to Aut(E[\ell^\infty])\cong GL_2(\Bbb{Z}_\ell)$ and $\iota$ is an embedding $\Bbb{Q}_\ell\to \Bbb{C}$ which gives an embedding $GL_2(\Bbb{Z}_\ell)\to GL_2(\Bbb{C})$. Then $\rho$ is a continuous representation so it suffices to know $\rho(Frob_L),\rho(g)$ because $\langle g,Frob_L\rangle$ is dense in $Gal(L^{unr}/K)$. But you can't do it for $\iota\circ \rho$ because it is a non-continuous representation. $\iota\circ \rho$ depends highly on $\ell$ and $\iota$ but $Tr\circ \iota\circ \rho$ doesn't, because it stays the same on the reduction where it becomes an isogeny thus it has integer trace.
The field extension $K(E[\ell^n])/K(E[\ell])$ is tamely ramified because it is Galois with Galois group a subgroup of $\{ M\in GL_2(\Bbb{Z/\ell^n Z}), M\equiv I\bmod \ell\}$ which has $\ell^{4(n-1)}$ elements which is coprime with $p$.
The good reduction assumption is not only there to ensure the correspondence of some $Frob_L$ with an isogeny of the reduction $\widetilde{E}$ whose traces are integers.
If $E$ has good reduction over $L=K(E[\ell^n])$ then the rational map encoding $P\to [\ell^m] P$ can be reduced $\bmod (\pi_L)$, it has $\ell^{2m}-1$ poles, so does its reduction which is separable too, thus Hensel lemma applies: the points of $\widetilde{E}[\ell^m]$ lift to $E[\ell^m]$, and the lift is a group isomorphism. And hence if the reduction of $\sigma \in G_{\overline{L}},\sigma\in Aut(E[\ell^\infty])$ acts trivially on $\widetilde{E}[\ell^\infty]$ then it will act trivially on $E[\ell^\infty]$. Whence $K(E[\ell^\infty])/L$ is unramified, ie. $L^{unr}=\bigcup_{p\ \nmid\ m} L(\zeta_m)$ contains $K(E[\ell^\infty])$.