Query about elliptic integral of third kind

Dear colleagues, I have the following problem: consider the function

F(z) = \int_0 ^z \sqrt{ (t-z) / (t (t-1) (t-2) )} dt

(the zeros 0, 1, and 2 of the denominator are chosen for simplicity).
Observe that this is an elliptic integral, and parameter $z$ is both
in the upper limit of integration and in the numerator of the square
root. What are the properties of F(z) as an analytic function on the
plane? What is its Riemann surface? In particular, how many solutions
an equation
F(z) = y
can have? It looks such a classical problem, but I cannot figure out
how to tackle it. So, any advice is most appreciated (a reference, an
idea, etc.).

Thanks in advance,
Andrei.

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