Ramification at infinity of a superelliptic covering ($w^r=z^n-1,d=\gcd (n,r),e_{\infty }=r/d$)

In the compactified Riemann surface of $w^r=z^n-1$, a positive circuit enclosing the $n$ simple finite zeros sends the sheet label $j$ to $j+n$ modulo $r$. This permutation has $d=\gcd (n,r)$ cycles of length $r/d$. Each cycle fills in one point over infinity, with ramification index $r/d$. Hence infinity is unramified exactly when $r$ divides $n$. Each finite zero has ramification index $r$, so the base sphere has exactly $n$ branch values in that case and $n+1$ otherwise.