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  "body": "\n## Summary\n\nThe $j$-invariant is the function $j = 1728\\, E_4^3 / (E_4^3 - E_6^2)$ on the [[upper half-plane|X6XGGH5FPW]], built from [[Eisenstein series|EYREVY31BN]].\nIt is invariant under the [[modular group|VEKV1EM77B]], holomorphic on $\\mathbb{H}$, and has a simple pole at infinity.\nIt gives a bijection from $\\mathrm{SL}_2(\\Z) \\backslash \\mathbb{H}$ onto $\\C$.\nTwo [[elliptic curves|427N6S6QQW]] over $\\C$ are isomorphic exactly when they have the same $j$-invariant.\n\n## Explanations\n\n### L1 human\n\nEach point $\\tau$ of the upper half-plane describes a lattice in the plane and so an elliptic curve.\nDifferent $\\tau$ can give the same curve, namely when they differ by the modular group.\nThe $j$-invariant is a single complex number that tells exactly which curve one has.\nIts expansion $j = q^{-1} + 744 + 196884 q + \\cdots$ has coefficients linked to the Monster group by [[monstrous moonshine|E0Y7Y75T6C]].\n",
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