= Solution
For a nonconstant <holomorphic map> $f:X\to Y$ between compact connected <Riemann surfaces>, its <degree of a holomorphic map>[degree] is
$$
\deg f=\sum_{x\in f^{-1}(y)}e_x,
$$
where $e_x$ is the <ramification index of a holomorphic map> at $x$. The <valency theorem> makes this sum independent of $y\in Y$. The <Riemann-Hurwitz formula> is
$$
\boxed{2g_X-2=(\deg f)(2g_Y-2)+\sum_{x\in X}(e_x-1).}
$$
Write $E=\mathbb C/\Lambda$. The map $\psi(z+\Lambda)=-z+\Lambda$ is well-defined and <biholomorphic>, with $\psi^2=\operatorname{id}$. Its fixed points satisfy
$$
z+\Lambda=-z+\Lambda
\iff2z\in\Lambda.
$$
They are therefore indexed by $\tfrac12\Lambda/\Lambda\cong(\mathbb Z/2\mathbb Z)^2$, so there are four, as in the <negation map on a one-dimensional complex torus>.
For each $x\in E'$ choose a small open disc $U_x$ for which $U_x\cap\psi(U_x)=\varnothing$. The projection restricts to a homeomorphism
$$
p|_{U_x}:U_x\longrightarrow p(U_x).
$$
Use $(p|_{U_x})^{-1}$ followed by a holomorphic coordinate on $U_x$ as a chart on $p(U_x)$. Choosing $\psi(U_x)$ instead changes this coordinate by the holomorphic map induced by $\psi$, so the transition functions are holomorphic. These charts make $S'$ a <Riemann surface> and $p:E'\to S'$ a holomorphic double <covering map>.
At each omitted fixed point, a centred local coordinate changes by $z\mapsto-z$, so the invariant quotient coordinate is $w=z^2$. Hence the extension has degree two and exactly four ramification points, each with $e_x=2$. Since the <complex torus> $E$ has genus one, <Riemann-Hurwitz formula>[Riemann--Hurwitz] gives
$$
0=2(2g_S-2)+4.
$$
Thus \b[$\boxed{g_S=0}$], in agreement with the general <quotient of a one-dimensional complex torus by negation>.
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