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.

**Knowl status:**

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

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Not referenced anywhere at the moment.

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- 2018-06-18 04:35:54 by John Jones (Reviewed)