An elliptic curve whose endomorphism ring is larger than \(\Z\) is said to have **complex multiplication** (often abbreviated to CM). In this case, for curves defined over fields of characteristic zero, the endomorphism ring is isomorphic to an order in an imaginary quadratic field. The discriminant of this order is the **CM discriminant**.

An elliptic curve whose geometric endomorphism ring is larger than \(\Z\) is said to have **potential complex multiplication** (potential CM). In the literature, these too are often called CM elliptic curves.

The property of having potential CM depends only on the $j$-invariant of the curve. In characteristic $0$, CM $j$-invariants are algebraic integers, and there are only finitely many in any given number field. There are precisely 13 CM $j$-invariants in $\Q$ (all integers), associated to the 13 imaginary quadratic orders of class number $1$:

$$ \begin{array}{c|ccccccccccccc} j & -12288000 & 54000 & 0 & 287496 & 1728 & 16581375 & -3375 & 8000 & -32768 & -884736 & -884736000 & -147197952000 & -262537412640768000\\ \text{CM discriminant} &-27 & -12 & -3 & -16 & -4 & -28 & -7 & -8 & -11 & -19 & -43 & -67 & -163 \end{array} $$

CM elliptic curves are examples of CM abelian varieties.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Bjorn Poonen on 2022-03-26 15:03:17

**Referred to by:**

- dq.ec.reliability
- dq.ec.source
- dq.ecnf.source
- ec.curve_label
- ec.endomorphism_ring
- ec.galois_rep_adelic_image
- ec.galois_rep_elladic_image
- ec.maximal_elladic_galois_rep
- ec.maximal_galois_rep
- ec.q.121.b2.bottom
- ec.q.invariants
- ec.q_curve
- g2c.geom_end_alg
- modcurve.cm_discriminants
- rcs.rigor.ec.q
- rcs.source.ec
- rcs.source.ec.q
- st_group.1.2.A.1.1a.top
- st_group.1.2.B.1.1a.bottom
- lmfdb/ecnf/main.py (lines 450-452)
- lmfdb/ecnf/main.py (line 833)
- lmfdb/ecnf/main.py (line 840)
- lmfdb/ecnf/templates/ecnf-curve.html (line 144)
- lmfdb/ecnf/templates/ecnf-curve.html (lines 152-154)
- lmfdb/elliptic_curves/elliptic_curve.py (line 429)
- lmfdb/elliptic_curves/elliptic_curve.py (line 1138)
- lmfdb/elliptic_curves/templates/ec-curve.html (lines 189-191)
- lmfdb/elliptic_curves/templates/ec-isoclass.html (line 34)
- lmfdb/elliptic_curves/templates/ec-isoclass.html (line 110)

**History:**(expand/hide all)

- 2022-03-26 15:03:17 by Bjorn Poonen (Reviewed)
- 2022-03-26 14:52:52 by Bjorn Poonen
- 2020-12-01 07:07:50 by John Cremona (Reviewed)
- 2020-12-01 06:57:59 by John Cremona
- 2020-10-10 16:19:37 by Andrew Sutherland (Reviewed)
- 2020-10-10 12:54:14 by Andrew Sutherland
- 2020-10-04 10:53:39 by John Cremona
- 2020-09-26 17:08:08 by John Voight
- 2020-09-26 17:07:35 by John Voight
- 2018-06-18 02:42:39 by John Jones (Reviewed)

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