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  "body": "\n## Summary\n\nAn [[elliptic curve|427N6S6QQW]] has complex multiplication when it has more [[endomorphisms|0J7H0YBFDJ]] than the multiplication-by-$n$ maps.\nIn characteristic zero its endomorphism ring is then an order in an imaginary [[quadratic field|7SV4WA92SM]].\nThe theory of complex multiplication describes the [[j-invariants|FEC2XH822R]] and torsion points of such curves.\nThese values generate abelian extensions of imaginary quadratic fields, such as the [[Hilbert class field|WWCJ7CR2R3]].\n\n## Explanations\n\n### L1 human\n\nOn every elliptic curve one can add a point to itself $n$ times, which gives an endomorphism \"multiplication by $n$\".\nSome special curves have extra symmetries.\nFor example, on $y^2 = x^3 - x$ the map $(x, y) \\mapsto (-x, iy)$ sends the curve to itself and respects addition, and doing it twice gives $P \\mapsto -P$.\nSo this map behaves like multiplication by the complex number $i$, hence the name.\nThese special curves are rare, but they have a particularly rich arithmetic.\n",
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