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  "body": "\n## Summary\n\nAn elliptic curve over a [[field|04C5ZBG1WT]] $K$ is a smooth projective curve of genus one over $K$ together with a chosen point defined over $K$.\nAway from characteristics $2$ and $3$, it is given by an equation $y^2 = x^3 + ax + b$ with $4a^3 + 27b^2 \\ne 0$.\nIts points form an abelian group under the [[chord-and-tangent law|P7V1BQJPSS]], which makes it the simplest abelian variety.\nElliptic curves are central in number theory, for example in the proof of Fermat's Last Theorem, and in cryptography.\n\n## Explanations\n\n### L1 human\n\nOver the real numbers, an elliptic curve looks like a smooth curve in the plane, in one or two pieces, symmetric about the $x$-axis.\nIts key feature is that one can \"add\" two points on it: draw the line through them, take the third intersection point, and reflect it in the $x$-axis.\nFor example, $y^2 = x^3 - x$ is an elliptic curve, since $4 \\cdot (-1)^3 + 27 \\cdot 0^2 = -4 \\ne 0$.\nThe curve $y^2 = x^3$ is not one, because it has a sharp cusp at the origin.\nDespite the name, an elliptic curve is not an ellipse; the name comes from elliptic integrals.\n",
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