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  "body": "\n## Definition\n\nLet $K$ be a field, and let $E$ be an elliptic curve over $K$.\nLet $y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$ be a [[Weierstrass equation|Z3C8B6QTS6]] of $E$, with quantities $b_2, b_4$ and discriminant $\\Delta \\ne 0$.\nPut $c_4 = b_2^2 - 24 b_4$.\nThe *$j$-invariant* of $E$ is\n$$j(E) = \\frac{c_4^3}{\\Delta} \\in K.$$\n\n## Remarks\n\nIf the characteristic of $K$ is not $2$ or $3$ and $E$ is $y^2 = x^3 + Ax + B$, then $c_4 = -48A$ and $\\Delta = -16(4A^3 + 27B^2)$, so\n$$j(E) = 1728 \\, \\frac{4A^3}{4A^3 + 27B^2}.$$\nThe value does not depend on the chosen Weierstrass equation of $E$.\nOver an [[algebraically closed field|0PF6TM8TWT]], two [[elliptic curves|427N6S6QQW]] are isomorphic if and only if they have the same $j$-invariant.\nMoreover, every element of the field is the $j$-invariant of some elliptic curve.\nExample: $y^2 = x^3 - x$ has $A = -1$ and $B = 0$, so $j = 1728 \\cdot (-4)/(-4) = 1728$.\nNon-example: $y^2 = x^3$ has $\\Delta = 0$, so it is not an elliptic curve and $c_4^3/\\Delta$ is not defined.\nOver a field that is not algebraically closed, curves with the same $j$-invariant can fail to be isomorphic: $y^2 = x^3 - x$ and $y^2 = x^3 + x$ both have $j = 1728$ but are not isomorphic over $\\Q$.\n",
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