Counting of Holomorphic Orbi-spheres in ${\mathbb{P}}_{2,2,2,2}^1$ and Determinant Equation

We count the number of holomorphic orbi-spheres in the ${\mathbb{Z}}_2$-quotient of an elliptic curve. We first prove that there is an explicit correspondence between the holomorphic orbi-spheres and the sublattices of $\mathbb{Z}\oplus \mathbb{Z}\sqrt{-1}(\subset \mathbb{C})$. The problem of counting sublattices of index $d$ then reduces to find the number of integer solutions of the equation $\alpha \delta -\beta \gamma =d$ up to an equivalence.