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  "body": "\n## Definition\n\nLet $K$ be a [[field|04C5ZBG1WT]].\nA *Weierstrass equation* over $K$ is an equation\n$$y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$$\nwith $a_1, a_2, a_3, a_4, a_6 \\in K$.\nIts associated quantities are\n$$b_2 = a_1^2 + 4a_2, \\quad b_4 = 2a_4 + a_1 a_3, \\quad b_6 = a_3^2 + 4a_6,$$\n$$b_8 = a_1^2 a_6 + 4a_2 a_6 - a_1 a_3 a_4 + a_2 a_3^2 - a_4^2,$$\nand its *discriminant* is\n$$\\Delta = -b_2^2 b_8 - 8 b_4^3 - 27 b_6^2 + 9 b_2 b_4 b_6.$$\n\n## Remarks\n\nThe projective closure of the curve in the [[projective plane|ZT9A6JK8QF]] adds one point $O = [0 : 1 : 0]$.\nThis closure is smooth exactly when $\\Delta \\ne 0$, and then it is an [[elliptic curve|427N6S6QQW]] with origin $O$.\nIf the characteristic of $K$ is not $2$ or $3$, a linear change of variables brings the equation to the short form $y^2 = x^3 + Ax + B$, for which $\\Delta = -16(4A^3 + 27B^2)$.\nExample: $y^2 = x^3 - x$ has $A = -1$ and $B = 0$, so $\\Delta = -16 \\cdot (-4) = 64 \\ne 0$, and it defines an elliptic curve over $\\Q$.\nNon-example: $y^2 = x^3$ is a Weierstrass equation with $\\Delta = 0$; the curve has a cusp at $(0, 0)$, where both partial derivatives of $y^2 - x^3$ vanish, so it is not an elliptic curve.\n",
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