I guess I need to explain my question. Let $C$ be a smooth genus $2$ curve, and let $J$ be its Jacobian. Fix an immersion $\varphi: C\to J$ and we identify $C$ with its image $\varphi(C)$. Now consider the two maps (1). Let $P\in J$ be a point of order $p$ and $\langle P \rangle$ the cyclic subgroup generated by $P$, and let $f:J\to J/\langle P \rangle$. (2). Let $\iota: J\to J: Q\mapsto -Q$ to be the involution. And let $g: J\to J/\langle \iota \rangle$.
My question is: How do we compute the (geometric) genus of $f(C)$ and $g(C)$?
I ask this question since I am reading the paper Density of rational points on elliptic K3 surfaces, and I am stuck on the proof of Lemma 2.11.
Thanks for your help.