Birational Maps of X(1) into P2

In this paper we study birational maps of modular curve X(1) attached to SL2(Z) into the projective plain P2. We prove that every curve of genus 0 and degree q in P2 can be uniformized by modular forms for SL2(Z) of weight 12q but not with modular forms of smaller weight, and that the corresponding uniformization can be chosen to be a birational equivalence. We study other regular maps X(1) → P2 and we compute the equation of obtained projective curve. We provide numerical examples in SAGE.