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It is well-known, that the moduli space $\mathcal M_{1;1}$ of elliptic curves is isomorphic to an orbifold space $(S_3\times S_2) \backslash\backslash \mathcal M_{0;4}$, where the first factor of the group acts by permutation of the first three distinguished points on a rational curve, and the second one acts trivially. Let us try to exploit this idea to construct $\mathcal M_{1;2}$.

Note first, that Riemann-Roch implies that for any two (different) points $P$ and $Q$ on a genus 1 curve, there exists a degree 2 function having degree 1 poles in $P$ and $Q$. The set of critical points is the same for any such a function. It is not difficult to understand, that the set of critical values of any such a function is the same up to an automorphism of $P^1$ fixing the infinity.

Vise-versa, if one fixes four points on $\mathbb C\subset P^1$, it determines a genus one curve with two distinguished points coming as preimages of the infinity under a degree two function ramified in the chosen points.

That indicates, that there should exist an orbifold map $\mathcal M_{1;2} \to (S_4) \backslash\backslash \mathcal M_{0;5}$, that forgets the numbering of the distinguished points on a genus 1 curve. But from the other hand, any genus 1 curve with two fixed points admits an automorphism interchanging the fixed points.

So my question is the following: is the map $\mathcal M_{1;2} \to (S_4) \backslash\backslash \mathcal M_{0;5}$ an isomorphism?

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Yes, it is.

See Leila Schneps, Special loci in moduli spaces of curves (in Galois Groups and Fundamental Groups, MSRI series 41, Cambridge University Press, 2003), pages 34-35.

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