---
tag: FEC2XH822R
kind: concept
name: j-invariant
msc: [11F03, 14H52]
sources:
  - url: https://en.wikipedia.org/wiki/J-invariant
same_as:
  wikidata: Q287419
  wikipedia: J-invariant
---

## Summary

The $j$-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]].
It is invariant under the [[modular group|VEKV1EM77B]], holomorphic on $\mathbb{H}$, and has a simple pole at infinity.
It gives a bijection from $\mathrm{SL}_2(\Z) \backslash \mathbb{H}$ onto $\C$.
Two [[elliptic curves|427N6S6QQW]] over $\C$ are isomorphic exactly when they have the same $j$-invariant.

## Explanations

### L1 human

Each point $\tau$ of the upper half-plane describes a lattice in the plane and so an elliptic curve.
Different $\tau$ can give the same curve, namely when they differ by the modular group.
The $j$-invariant is a single complex number that tells exactly which curve one has.
Its expansion $j = q^{-1} + 744 + 196884 q + \cdots$ has coefficients linked to the Monster group by [[monstrous moonshine|E0Y7Y75T6C]].