All varieties are assumed to be projective over $\mathbb{C}$. Let $f\_1: Y \to X$ and $f\_2: Y' \to X$ be étale morphisms with same finite Galois groups (to be honest, I don't know what does Galois group really mean in this context), then can we conclude $Y,Y'$ are birational?


Even if the general situation might have negative answer, are there any results concerning problemss along this line?



Étale covers and birationality of varieties
I think, for an elliptic curve $E/\mathbf{C}$, there are étale coverings $E\_i \to E$, $i = 1,2$ of degree a prime number (hence with the same Galois group; the étale fundamental group of an elliptic curve is Abelian), with $E\_1 \not\cong E\_2$.


