I am currently reading through the book An Introduction to Teichmüller Spaces by Imayoshi and Taniguchi. In Section 1.2, we see that $M_1$, the moduli space of tori, can be identified with $\mathbb{H}/PSL(2,\mathbb{Z})$. This is clear, since two tori $R_\tau$ and $R_{\tau'}$ - generated by normalized lattices having sides $1, \tau$ and $1, \tau'$, respectively - are biholomorphically equivalent if and only if $\tau=\tau'$ (where $\tau \in \mathbb{H}$), and in particular we can identify this with $\mathbb{C}$ using the $j$-invariant of an elliptic curve (associated to a torus generated by the normalized lattice with sides $1, \tau$).
We also know that for a given cross ratio $\{z_1, z_2, z_3, z_4\} = \lambda$, permutation of the $z_i$ by an element of $S_4$ results in the cross ratio being one of the following: $\lambda, \frac{1}{\lambda}, 1-\lambda, \frac{1}{1-\lambda}, \frac{\lambda-1}{\lambda}, \frac{\lambda}{\lambda-1}$, and we can put the 4-tuple having cross ratio $\lambda$ in "canonical" form by setting it to be $\{0, 1, \lambda, \infty\}$. Next, a torus $S_\lambda$ is given by the following equation (depending on $\lambda$): $w^2 = z(z-1)(z-\lambda)$. Then two tori $S_\lambda$ and $S_{\lambda'}$ are biholomorphically equivalent if and only if there is a linear fractional transformation taking $\{0, 1, \lambda, \infty\}$ to $\{0, 1, \lambda', \infty\}$, so then $\lambda'$ can only be one of the values after a permutation as above.
Let $G$ be the group (which is actually just $S_3$) generated by the two functions $\lambda \mapsto \frac{1}{\lambda}$ and $\lambda \mapsto 1-\lambda$ (note these functions are analytic automorphisms of $D = \mathbb{C}-\{0, 1\}$). In particular we have the $S_4/V \approx S_3$, where $V$ is the Klein 4-group (and is in fact the permutations that fix the cross ratio). The book goes on to say that this shows $M_1 \approx D/G$ and there is a biholomorphic map $F: D/G \to \mathbb{C}$ given by $F([\lambda]) = f(\lambda) = \frac{{(\lambda^2-\lambda+1)}^3}{\lambda^2{(\lambda-1)}^2}$.
I do not understand where exactly $f(\lambda)$ comes from or how the identification is clear, but the setup makes sense to me. Does this function have a special name? Does it appear anywhere else (in a significant way)?