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  "body": "\n## Definition\n\nLet $K$ be a [[field|04C5ZBG1WT]].\nAn *elliptic curve over $K$* is a pair $(E, O)$ with two properties.\n$E$ is a smooth projective curve over $K$ of genus $1$, and $O \\in E(K)$ is a point of $E$ defined over $K$.\n\nEquivalently, it is a curve in the projective plane given by a Weierstrass equation with $a_1, \\ldots, a_6 \\in K$:\n$$y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.$$\nIts discriminant $\\Delta$ must be nonzero, and $O = [0 : 1 : 0]$ is the point at infinity.\nIf the characteristic of $K$ is not $2$ or $3$, every elliptic curve has an equation $y^2 = x^3 + ax + b$ with $a, b \\in K$, and the condition $\\Delta \\ne 0$ becomes $4a^3 + 27b^2 \\ne 0$.\n\n## Remarks\n\nExample: $E \\colon y^2 = x^3 - x$ over $\\Q$.\nHere $a = -1$, $b = 0$, and $4a^3 + 27b^2 = -4 \\ne 0$, so the cubic $x^3 - x = x(x-1)(x+1)$ has three distinct roots and $E$ is smooth.\n\nNon-example: $y^2 = x^3 + x^2$ has $\\Delta = 0$.\nIts projective closure has a node at $(0, 0)$, where both partial derivatives of $y^2 - x^3 - x^2$ vanish, so the smoothness condition fails.\n\nThe condition $4a^3 + 27b^2 \\ne 0$ says that $x^3 + ax + b$ has no repeated root in an algebraic closure of $K$.\nFor $y^2 = x^3 + ax + b$ one has $\\Delta = -16(4a^3 + 27b^2)$.\nThe set $E(K)$ of points over $K$, including $O$, is an abelian group with identity $O$ under the chord-and-tangent group law.\n",
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