Im(τ) range3
Re(τ) range1
Color modeArg
Terms N8
j(τ) = E₄(τ)³/Δ(τ)
= 1/q + 744 + 196884q + ...
= 1/q + 744 + 196884q + ...
The upper half-plane under SL(2,ℤ) transformations — the j-function assigns a complex number to every lattice, invariant under modular symmetry.
The j-invariant classifies elliptic curves over ℂ: two curves are isomorphic iff they share the same j-value. Monstrous moonshine (Conway-Norton, Borcherds 1998 Fields Medal): the coefficients of j are dimensions of Monster group representations.