I've been playing around with Riemann surfaces of cubics, and it seems to me that all surfaces obtained as 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.
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.
Is this correct? If so, what is the common J-invariant? And since we can choose $q(z)=z^3-1$, this means that all these curves have CM, right?