I've been playing around with Riemann surfaces of cubics, and it seems to me that all coverings of the Riemann sphere from equations of the form $w^3 = q(z)$, where $q(z)$ is a cubic with three distinct roots, must be isomorphic. Is this correct? (Argument given below.)

Mainly I want to know if there are any good references on this specific kind of cover.

Argument: we have a critical point of multiplicity 3 at each of the roots of $q(z)$. Monodromy around a small counterclockwise circuit about any of these points multiplies $w$ by the same cube root of unity ($\neq 1$). So monodromy around a circuit enclosing all three roots of $q(z)$ leaves $w$ unchanged, so no branch points over $\infty$.

Now we can move the three roots of $q(z)$ to any other positions using a Möbius transformation, so by the previous paragraph, we should be able to establish an analytic isomorphism between the surfaces via continuation.