Expected genus of a function field over a finite field

Question

Let $K=\mathbb F_q(t)$ be the field of rational functions over $\mathbb F_q$ and $f(T)\in K[T]$ be an irreducible polynomial. Let $F=K[y]/f(y)$ and $E$ be the Galois closure of F/K. What should I expect as the genus of $E$? I mean, if I "randomly" write a polynomial over a global function field K, this is very likely to be irreducible (since its Galois group is very likely the simmetric group), would it be possible to also say something similar about the genus of E? Any reference for such problem is very welcome.

Answer 1

I think it is difficult to say much without more information. Consider the following family of function fields. Let $q=2$ and $$ f(y)=y^2+y-t^{2m+1} $$ for some positive integer $m$. Then $F=K[y]/\langle f(y)\rangle$ is Galois over $K$ with Galois group $C_2$ generated by the $K$-automorphism $\sigma:y\mapsto y+1$. But we know that $F$ has genus $m$.

This implies that there is for example no upper bound on the genus as a function of the size of the Galois group.