Let $E$ be a compact Riemann surface of genus 1, i.e., an elliptic curve.
Let $P$ be the identity element of $E$.
Question 1. Does there exist a cofinite Fuchsian group (or a Fuchsian group of the first kind) $\Gamma$ such that $E-\{P\}=\Gamma\backslash\mathfrak{h}$, where $\mathfrak{h}$ denotes the complex upperhalf plane, and $E$ can be obtained by adding the ''cusp'' $P$?
Question 2. Does there exist a cofinite Fuchsian group $\Gamma$ without elliptic elements and a finite set of points $\{b_1,\ldots,b_n\}\subset E$ such that $E-\{b_1,\ldots,b_n\} =\Gamma\backslash\mathfrak{h}$ and $E$ can be obtained by adding the ''cusps'' $b_i$.
If yes, how are these groups related to $E$? What can be said about them?
I would like to choose $b_1,b_2,b_3$ and $b_4$ to be the ramification points of the projection onto the $x$-coordinate, i.e., Weierstrass P-function.