We count the number of holomorphic orbi-spheres in the Z₂-quotient of an elliptic curve. We first prove that there is an explicit correspondence between the holomorphic orbi-spheres and the sublattices of Z ⊕ Z \sqrt{-1} (⊂ C). The problem of counting sublattices of index d then reduces to find the number of integer solutions of the equation α δ - β γ = d up to an equivalence.

The work of H. Hong described in this paper was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK2130405 and CUHK2130446).