I'm looking for a reference of an isogeny fact that I've used many times but am having a hard time proving formally.

One can define the degree of an isogeny as the degree of extension fields of the function fields of two elliptic curves using the pullback.

One can also write an isogeny $\phi$ as a map $\phi((x,y)) = (f(x),yg(x))$, define $f(x) = p(x)/q(x)$ and then say that the degree of $\phi$ is equal to the maximum of the degree of $p$ and $q$.

My question is why/how are these two definitions equivalent? Is there a standard reference that includes this proof?

For an online reference of the question I'm posing, see Andrew Sutherland's course notes on page 7 at