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  "body": "\n## Definition\n\nLet $K$ be a field of characteristic $0$ with [[algebraic closure|0X6AB5P3G1]] $\\overline{K}$, and let $E$ be an [[elliptic curve|427N6S6QQW]] over $K$.\nLet $\\End(E)$ be the ring of [[endomorphisms|0J7H0YBFDJ]] of $E$ defined over $\\overline{K}$, that is, morphisms $E \\to E$ over $\\overline{K}$ that send the origin $O$ to $O$.\nFor $n \\in \\Z$ let $[n] \\colon E \\to E$ be multiplication by $n$; the map $n \\mapsto [n]$ embeds $\\Z$ in $\\End(E)$.\nThe curve $E$ has *complex multiplication*, or is a *CM elliptic curve*, if $\\End(E) \\ne \\Z$.\nThat is, $E$ has an endomorphism that is not of the form $[n]$.\n\n## Remarks\n\nIn characteristic $0$, $\\End(E)$ is either $\\Z$ or an order $\\mathcal{O}$ in an imaginary [[quadratic field|7SV4WA92SM]] $F$, and one then says that $E$ has CM by $\\mathcal{O}$.\nOver $\\C$, write $E \\cong \\C/\\Lambda$ for a lattice $\\Lambda$; then $\\End(E) \\cong \\set{\\alpha \\in \\C : \\alpha \\Lambda \\subseteq \\Lambda}$, which explains the name.\n\nExample: $E \\colon y^2 = x^3 - x$ over $\\Q$.\nThe map $[i](x, y) = (-x, iy)$ sends $E$ to itself, since $(iy)^2 = -(x^3 - x) = (-x)^3 - (-x)$.\nIt fixes $O$, and $[i] \\circ [i] = [-1]$, so $[i] \\ne [n]$ for every integer $n$; hence $E$ has CM, by $\\Z[i]$.\n\nNon-example: $y^2 = x^3 + x + 1$ over $\\Q$ has $j$-invariant $1728 \\cdot \\frac{4}{31} = \\frac{6912}{31}$, which is not an algebraic integer.\nThe $j$-invariant of a CM curve is always an algebraic integer, so this curve has endomorphism ring $\\Z$.\n\nOver a finite field the endomorphism ring of every elliptic curve is larger than $\\Z$, so the notion is mainly used in characteristic $0$.\n",
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