---
tag: 9538W1K1EX
kind: concept
name: j-invariant of an elliptic curve
msc: [14H52, 11G05]
sources:
  - url: https://en.wikipedia.org/wiki/J-invariant
---

## Summary

The $j$-invariant of an [[elliptic curve|427N6S6QQW]] is a number computed from the coefficients of a [[Weierstrass equation|Z3C8B6QTS6]] of the curve.
It does not change under changes of coordinates.
Over an algebraically closed field, two elliptic curves are isomorphic exactly when they have the same $j$-invariant.
So the $j$-invariant is a coordinate on the set of isomorphism classes of elliptic curves.
Over $\C$ it agrees with the value of the modular [[j-invariant|FEC2XH822R]] at the period ratio $\tau$ of the curve.

## Explanations

### L1 human

Many different equations describe the same elliptic curve after a change of variables.
The $j$-invariant is a single number that sees through all these changes.
For $y^2 = x^3 + ax + b$ it is $1728 \cdot 4a^3 / (4a^3 + 27b^2)$.
For instance, every curve $y^2 = x^3 + ax$ with $a \ne 0$ has $j = 1728$, and every curve $y^2 = x^3 + b$ with $b \ne 0$ has $j = 0$.