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  "body": "\n## Summary\n\nThe $j$-invariant of an [[elliptic curve|427N6S6QQW]] is a number computed from the coefficients of a [[Weierstrass equation|Z3C8B6QTS6]] of the curve.\nIt does not change under changes of coordinates.\nOver an algebraically closed field, two elliptic curves are isomorphic exactly when they have the same $j$-invariant.\nSo the $j$-invariant is a coordinate on the set of isomorphism classes of elliptic curves.\nOver $\\C$ it agrees with the value of the modular [[j-invariant|FEC2XH822R]] at the period ratio $\\tau$ of the curve.\n\n## Explanations\n\n### L1 human\n\nMany different equations describe the same elliptic curve after a change of variables.\nThe $j$-invariant is a single number that sees through all these changes.\nFor $y^2 = x^3 + ax + b$ it is $1728 \\cdot 4a^3 / (4a^3 + 27b^2)$.\nFor 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 $j = 0$.\n",
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