Let $ K $ be a $ p $-adic field. Suppose we have an isogeny of elliptic curves $ \phi : E \to E' $ defined over $ K $, where $ E $ and $ E' $ both have multiplicative reduction.

1) Is there anything we can say about the structure of the induced map on the Tate modules $ V_l(E) \to V_l(E') $? Mostly I'm interested in the eigenvalues.

I should note that for question 1) it is actually easy to determine the determinant of this linear map using the Weil pairing. I'm only interested in additional results, such as a way to compute the trace.

Note that by possibly enlarging the field K we may assume the multiplicative reduction for both curves is split, hence $ E $ and $ E' $ are isomorphic to Tate curves $ E_q $ and $ E_{q'} $ with $ q, q' \in K^{*} $ of positive valuation. Note that for any finite extension $ L / K $ we have that $ E_q( L ) \cong L^{*} / q^{\mathbb{Z}} $. For a suitably large choice of $ L $ the $ m $-torsion points of the latter have a basis $ \{ \zeta_m , q^{1/m} \} $, where $ \zeta_m $ is an $ m $-th root of unity.

2) Can we say anything about the structure of the induced map $ \phi : E[m] \to E'[m] $ with respect to the bases given by $ \{ \zeta_m, q^{1/m} \} $ and $ \{ \zeta_m, (q')^{1/m} \} $?

Although I'm interested in general results, I would be happy getting a result in case E' is a Galois conjugate of E, so in case q' is a Galois conjugate of q.