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The endomorphism ring $\End(E)$ of an elliptic curve $$E$$ over a field $K$ is the ring of all endomorphisms of $$E$$ defined over $K$. For endomorphisms defined over extensions, we speak of the geometric endomorphism ring of $E$.

For elliptic curves defined over fields of characteristic zero, this ring is isomorphic to $$\Z$$, unless the curve has complex multiplication (CM) defined over the ground field, 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.

$\End(E)$ always contains a subring isomorphic to $\Z$, since for $m\in\Z$ there is the multiplication-by-$m$ map $[m] \colon E\to E$.

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

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• Review status: reviewed
• Last edited by John Voight on 2020-09-26 17:00:04
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