I am interested in an example of the following situation, over an algebraically closed field o zero characteristic.
Let $E$ be an elliptic curve, and $G$ a finite group of automorphisms of $E$ (as an algebraic variety). $G$ induces an action on the function field $k(E)$. Let $C$ be an smooth projective curve with function field $k(E)^G$.
By Hurwitz formula, two situations can happen: either $C$ again has genus $1$, or $C=\mathbb{P}^1$.
Can someone show me an example of an elliptic curve and finite group of automorphisms such that $C$ is $\mathbb{P}^1$? More informally: an elliptic curve $E$ and finite group of automorphisms $G$ such that the quotient variety $E/G$ is rational.
Sure: the quotient of any elliptic curve by the involution $P\mapsto -P$ gives $\Bbb P^1$ on the nose. Using the standard Weierstrass equation, the quotient corresponds to projection to the $x$-axis. You may also compute the fixed field: again under the standard Weierstrass equation, the function field of your elliptic curve is $k(x,y)$ with $y^2=x^3+Ax+B$ for some $A,B\in\Bbb C$. Then the involution acts on the function field by $y\mapsto -y$, which has fixed field $k(x)$.