Let $L$ be a number field and $E,E'$ two elliptic curves defined over $L$. Suppose $\varphi\colon E\to E'$ is an isogeny defined over $L$. Let $\mathfrak p$ be a prime of bad reduction for $E,E'$. Then the nonsingular points of the reductions of $E$ and $E'$, which I'll call $\widetilde{E}_{ns}$ and $\widetilde{E'}_{ns}$, are group varieties, isomorphic either to the additive or to the multiplicative group over $L$.
My first question is: does $\varphi$ induce a (in general nontrivial) map of group varieties $\widetilde{\varphi}\colon\widetilde{E}_{ns}\to\widetilde{E'}_{ns}$ in a functorial way? Can one can write down rational maps for $\varphi$ with $L$-integer coefficients and then reduce them modulo $\mathfrak p$?
If the answer to the first question is yes and the reduction at $\mathfrak p$ is multiplicative, does the degree of the reduced map $\widetilde{\varphi}$ equal the degree of the reduction of the dual isogeny $\widehat{\varphi}\colon E'\to E$?
Thank you very much!