A 2-Sheeted Riemann surface, with 2 branch cuts has a genus 1. A ring torus also has a genus 1 (In fact, section 13.4 of John Terning's book, modern supersymmetry and dynamics and duality claims that by joining branch cuts on one sheet with branch cuts on other gives a torus. For example, look at page 5 of this pdf.
My question is, is there a conformal mapping which maps 2-sheeted Riemann surface with 2 branch cuts to a torus? Is $y^2=x(x−1)(x−a)$ the "mapping" (here the branch points of the Riemann surface are at $0$, $1$, and infinity)?
Note: If there is any more information needed to make the question clear please let me know, I will try (I am not from a mathematical background). As a physicist, I understand torus as a rectangle on a patch of complex plane with periodic boundary conditions (or a rectangle where the opposite sides are identified with each other.)
Also, does a torus have translational symmetry (In the sense that for example, any point z in the complex plane, is similar to $z + c$ where $c$ is some constant)?