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Jun 29, 2016 at 8:59 comment added rita Since $\phi$ is a double cover, you have $K_C=\phi^*L$ for some $L\in Pic(E)$. It follows that $2P$ is (up to linear equivalence) a pull back from $E$. Since $h^0(C, 2P)=1$, the divisor $2P$ is invariant under the elliptic involution, so $P$ is a ramification point of $\phi$. Maybe this helps.
Jun 28, 2016 at 15:43 review First posts
Jun 28, 2016 at 15:53
Jun 28, 2016 at 15:40 history asked M. Jones CC BY-SA 3.0