Elliptic Curves with Everywhere Good Reduction over Quadratic Fields

Elliptic Curves with Everywhere Good Reduction over Quadratic Fields
compiled by
J. E. Cremona
University of Warwick, U.K.

Updated 2005-09-29 (with minor correction 2017-02-21)

Note that this table is no longer being updated, as it has been superceded by the LMFDB, which as of February 2017 contains 497 elliptic curves with everywhere good reduction.

We give tables showing the existence or non-existence of elliptic curves with everywhere good reduction defined over quadratic number fields Q(√d). This data is currently incomplete and will be updated. Where possible we give attributions and references: please email me if you see any errors or omissions.

Where we list no curves, we distinguish between cases where none are known but it is not proved that none exist from cases where it has been proved that non exist. Similarly when some curves are known to exist, we indicate whether or not the curves listed are known to be a complete list.

There are separate tables for real and imaginary quadratic fields. At present the tables give data for real fields Q(√d) for 1<d<100 and data for imaginary fields Q(√d) with 0>d>-100. For curve details (where available) follow the link on the value of d.

Real quadratic fields

d	# j-invariants	# isogeny classes	# isomorphism classes	References
2	0	0	0	[Kid99]
3	0	0	0	[Kid99]
5	0	0	0	[Pin84],[Kid99]
6	3	1	6	[Com90]
7	4	1	8	[Com90]
10	0	0	0	[Kid01abc]
11	0	0	0	[Kag05]
13	0	0	0	[Ish86], [KK97]
14	2	1	4	[Com90]
15	0	0	0	[Kid01abc]
17	0	0	0	[KK97]
19	0	0	0	[Kag05]
21	0	0	0	[KK97]
22	1	1	2	[Kid01abc]
23	0	0	0	[Kag05]
26	2	1	4	[Kid01abc]
29	4	1	4	[Kid96]
30	0	0	0	[Kid01abc]
31	0	0	0	[Kag05]
33	3	1	6	[Kid97]
34	0	0	0	[Kid01]
35	0	0	0	[Kid01]
37	2	1	2	[Kid96],[Kag98]
38	>0	>0	>0	[Cre07]
39	0	0	0	[Kid01]
41	>0	>0	>0	[Com90]
42	0	0	0	[Kid01]
43
46
47	0	0	0	[Kid01]
51
53	0	0	0	[Kag05]
55
57	0	0	0	[Kag01]
58	0	0	0	[Kid01]
59
61	0	0	0	[Kag05]
62
65	>0	>0	>0	[Com90]
66	0	0	0	[Kid01]
67
69	0	0	0	[Kag01]
70	0	0	0	[Kid01]
71
73	0	0	0	[KK97]
74	0	0	0	[Kid01]
77	2	1	4	[Kid01abc]
78
79	>0	>0	>0	[Kid01abc]
82	0	0	0	[Kid01]
83	0	0	0	[Kag05]
85	0	0	0	[Kid01]
86	>0	>0	>0	[Kid01abc]
87
89	0	0	0	[Kag05]
91
93	0	0	0	[Kag01]
94	0	0	0	[Kid01]
95
97	0	0	0	[KK97]

Bibliography

[CN87]	Salvador Comalada and Enric Nart. Courbes elliptiques avec bonne réduction partout. C. R. Acad. Sci. Paris Sér. I Math., 305(6):223-224, 1987.
[Com90]	Salvador Comalada. Elliptic curves with trivial conductor over quadratic fields. Pacific J. Math., 144(2):237-258, 1990.
[CN92]	Salvador Comalada and Enric Nart. Modular invariant and good reduction of elliptic curves. Math. Ann., 293(2):331-342, 1992.
[Cre07]	J. E. Cremona. Finding all elliptic curves with good reduction outside a given set of primes. Experimental Mathematics 16 No.3 (2007), 303-312.
[Ish86]	Hidenori Ishii. The nonexistence of elliptic curves with everywhere good reduction over certain quadratic fields. Japan. J. Math. (N.S.), 12(1):45-52, 1986.
[Kag01]	Takaaki Kagawa. Determination of elliptic curves with everywhere good reduction over real quadratic fields `Q(√(3p))`. Acta Arith., 96(3):231-245, 2001.
[Kag98]	Takaaki Kagawa. Determination of elliptic curves with everywhere good reduction over Q(√37). Acta Arith. LXXXIII.3 (1998).
[Kag05]	Takaaki Kagawa. Determination of elliptic curves with everywhere good reduction over real quadratic fields, II . preprint 2005.
[Kid96]	Masanari Kida. On a characterization of Shimura's elliptic curve over `Q(√37)`. Acta Arith., 77(2):157-171, 1996.
[KK97]	Masanari Kida and Takaaki Kagawa. Nonexistence of elliptic curves with good reduction everywhere over real quadratic fields. J. Number Theory, 66(2):201-210, 1997.
[Kid99]	Masanari Kida. Reduction of elliptic curves over certain real quadratic number fields. Math. Comp., 68(228):1679-1685, 1999.
[Kid00]	Masanari Kida. Computing elliptic curves having good reduction everywhere over quadratic fields. II. In Algebraic number theory and Diophantine analysis (Graz, 1998), pages 239-247. de Gruyter, Berlin, 2000.
[Kid01a]	Masanari Kida. Computing elliptic curves having good reduction everywhere over quadratic fields. Tokyo J. Math., 24(2):545-558, 2001.
[Kid01b]	Masanari Kida. Good reduction of elliptic curves over imaginary quadratic fields. J. Théor. Nombres Bordeaux, 13(1):201-209, 2001. 21st Journées Arithmétiques (Rome, 2001).
[Kid01c]	Masanari Kida. Nonexistence of elliptic curves having good reduction everywhere over certain quadratic fields. Arch. Math. (Basel), 76(6):436-440, 2001.

email: John.Cremona@gmail.com