Let $E/K, E'/K$ be elliptic curves defined over a number field $K$. Let $\phi: E \to E'$ be a non-constant isogeny defined over $K$. Why must the conductors be equal?
I know that this is an exercise in Chapter IV of Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, but I haven't yet carefully read through that chapter. I'm not looking for a technical answer, but for some intuition as to why this makes sense.