The moduli space M_g is a space whose points parametrize the isomorphism classes of algebraic curves (Riemann surfaces) of genus g. For genus 1 (elliptic curves / tori), M_1 ≅ ℍ/SL(2,ℤ) — the upper half-plane modulo the action of the modular group. A complex torus is ℂ/(ℤ + τℤ) for some τ with Im(τ) > 0, and two tori are isomorphic iff τ and τ' are related by a Möbius transformation τ → (aτ+b)/(cτ+d) with integer coefficients and determinant 1. The fundamental domain of this action (shown) is the famous modular curve. Varying τ sweeps through all elliptic curve shapes — the j-invariant j(τ) = 1728 · (4a³)/(4a³+27b²) classifies them completely.