I have read the book the Arithmetic of Elliptic curves of J.H Silverman (we can read in http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf)
In chapter 3, the proof of proposition 1.5
I don't understand why the map they consider $\phi:E\rightarrow \mathbb{P}^1$ like above has degree one. By the difinition of $\deg$ of mmorphism, we must find the degree of filed extension $[K(E):\phi^*\left(K(\mathbb{P}^1\right)]$, whereas $K(C)$ is the function field of $C$ over an algebraic closure field and $\phi^*:K(\mathbb{P}^1)\rightarrow K(E), f\mapsto f\circ \phi$. I can't imaginary what $\phi^*(K(\mathbb{P}^1))$ is. Somebody can help me?