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The endomorphism ring of an elliptic curve $$E$$ is the ring of all endomorphisms of $$E$$ (including those defined over extensions of the base field of $$E$$). For elliptic curves defined over $$\Q$$ and other fields of characteristic zero, this ring is isomorphic to $$\Z$$, unless the curve has complex multiplication (CM), in which case the endomorphism ring is an order in an imaginary quadratic field. For curves defined over $$\Q$$ this order is one of the 13 orders of class number one.

This is a special case of the endomorphism ring of an abelian variety.

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• Review status: reviewed
• Last edited by John Jones on 2018-06-18 04:35:54
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