Let $C : \text{"}y^2=f(x), z^2 = h(x)\text{"}$, $\text{deg}(f) = \text{deg}(g) = 3$, $f$ and $g$ without common roots, be a (affine) curve (with coordinates $[x:y:z:t]$ in the associated smooth projective curve, with I denote aswell by $C$). First, I want to compute its genus, and second, I want to compute its jacobian as the product of jacobian of three curves. Let $E_1$ the elliptic curve given in (an affine chart) by $y^2=f(x)$ and $E_2$ the elliptic curve given in (an affine chart) by $z^2 = h(x)$. I have a degree two cover : $$\phi_1 :\begin{array}~C \longrightarrow E_1 \\ [x:y:z:t] \mapsto [x:y:t] \end{array}$$
(in affine chart, we send $(x,y,z)$ to $(x,y)$)
The ramified finite point seems to be the roots of $h$, so we have $3$ ramified finite point of ramification index $2$. We have aswell a maximum of $2$ infinity point on $C$ (because of the degree of $\phi_1$), and by the Riemann-hurwitz formula, we have : $g(C) = 1+\frac{\sum_{Q \in C} e_{Q}-1}{2} \in \mathbb{N}$, so we must have only one point in infinity on $C$ which is ramified of index of ramification $2$, and finally $g(C) = 3$.
First problem : I've got my reason to believe we should have $g(C)=4$. Is there something wrong in what I've written above ?
About the jacobian : we already have two maps (given by the pullback) $\phi_{1}^* : E_1 \longrightarrow \text{Jac}(C)$ and $\phi_{2}^* : E_1 \longrightarrow \text{Jac}(C)$. It seems quite natural to look at the hyperelliptic curve (of genus 2) $F$ given by $w^2 = f(x)g(x)$ and we have a natural morphism of curve : $\phi_3 : [x:y:z:t] \mapsto [x:yz:t]$ from $C$ to $F$, and then a natural map : $\phi_{3}^* : \text{Jac}(F) \longrightarrow \text{Jac}(C)$. Now, if $g(C) = 4$, we would have $\text{dim}(\text{Jac}(C)) = \dim(E_1 \times E_2 \times \text{Jac}(F))$ and then we could maybe try to show that the map : $(D_1, D_2, D_3) \mapsto \sum_i \phi_{i}^*(D_i)$ from $E_1 \times E_2 \times \text{Jac}(F)$ to $\text{Jac}(C)$ is an isogeny. To do that, I thought about looking at the map induced on the global $1$-differential form (= tangent space at $0$ of the jacobian of the product) of $E_1 \times E_2 \times F$ and $C$ (induced by the pullback) and hope that it's surjective (i.e maybe look at the "basis" of $H^0(E_i, \Omega^1[E_i])$ and $H^0(E_i, \Omega^1[F])$, then pulling them back and hope we find $4$ linearly independant forms). Is that the right way to do ?
My problem is, still, I didn't find $g(C)=4$ but $g(C)=3$, and I suspect I made a mistake somewhere to compute the genus.
Someone could help me, please ?
Thank you !