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  "body": "\n## Summary\n\nThe points of an [[elliptic curve|427N6S6QQW]] $E$ over a field $K$, together with the point at infinity $O$, form an [[abelian group|082GHNPJ53]] $E(K)$.\nThe operation is the chord-and-tangent law: three points add up to $O$ exactly when they lie on a line.\nThis group structure is what makes elliptic curves so useful, from [[Mordell's theorem|BZHGKN6Q3K]] to elliptic curve cryptography.\n\n## Explanations\n\n### L1 human\n\nTo add two points $P$ and $Q$ on the curve, draw the line through them.\nIt meets the cubic curve in a third point; reflect that point in the $x$-axis to get $P + Q$.\nTo double a point, use the tangent line at that point instead.\nThe point at infinity plays the role of zero, and the reflection of a point is its negative.\nFor example, on $y^2 = x^3 + 1$ the tangent at $(2, 3)$ has slope $2$, and doubling gives $2 \\cdot (2, 3) = (0, 1)$.\n",
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