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.
- Review status: reviewed
- Last edited by Bjorn Poonen on 2022-03-26 15:03:17
- dq.ec.source
- dq.ecnf.source
- ec.curve_label
- ec.endomorphism_ring
- ec.galois_rep
- 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.minimal_twist
- ec.q_curve
- g2c.geom_end_alg
- modcurve.cm_discriminants
- modcurve.known_points
- 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 374-376)
- lmfdb/ecnf/main.py (line 756)
- lmfdb/ecnf/main.py (line 763)
- lmfdb/ecnf/templates/ecnf-curve.html (line 224)
- lmfdb/ecnf/templates/ecnf-curve.html (lines 236-238)
- lmfdb/elliptic_curves/elliptic_curve.py (line 441)
- lmfdb/elliptic_curves/elliptic_curve.py (line 1332)
- lmfdb/elliptic_curves/templates/bhkssw.html (line 92)
- lmfdb/elliptic_curves/templates/ec-curve.html (lines 211-213)
- lmfdb/elliptic_curves/templates/ec-isoclass.html (line 34)
- lmfdb/elliptic_curves/templates/ec-isoclass.html (line 110)
- lmfdb/modular_curves/main.py (line 1032)
- lmfdb/modular_curves/main.py (line 1155)
- lmfdb/modular_curves/templates/modcurve.html (line 203)
- lmfdb/modular_curves/templates/modcurve.html (line 242)
- 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)