Imaginary quadratic number fields with small exponent - Computer Algebra and Number Theory

Imaginary quadratic number fields with small exponent.

Here are lists with imaginary quadratic fields with exponent of the ideal class group up to 8. For exponent 1 the list is complete. For exponents 2,4,8 the lists are complete assuming that there are no Siegel zeros. Without this assumption there might be one missing field. For exponents 3 and 5 the lists are complete assuming the generalized Riemann hypothesis. Without this assumption it is known that there are finitely many fields with exponent 3 or 5. For exponent 6 the list is complete up to discriminant 3.1·10²⁰. It is known that there are finitely many fields with exponent 6. For exponent 7 the list is complete up to discriminant 3.1·10^20. It is not known whether there are only finitely many fields with exponent 7. Assuming the generalized Riemann hypothesis the exponent tends to infinity, hence there should be only finitely many fields with given exponent.

The scientific content is given in the paper:

A.-S. Elsenhans, J. Klüners, F. Nicolae, Imaginary quadratic number fields with class groups of small exponent , Acta Arith., 193, 2020, 217-233.

List of 9 fields with exponent 1

The following table contains all fields with exponent 1.

Discriminant	Factorization	Class group
-3	-3	C1
-4	-2^2	C1
-7	-7	C1
-8	-2^3	C1
-11	-11	C1
-19	-19	C1
-43	-43	C1
-67	-67	C1
-163	-163	C1

List of 56 fields with exponent 2

The following table is complete assuming that there are no Siegel zeros. Without any assumption there might be one missing field.

Discriminant	Factorization	Class group
-15	-3 * 5	C2
-20	-2^2 * 5	C2
-24	-2^3 * 3	C2
-35	-5 * 7	C2
-40	-2^3 * 5	C2
-51	-3 * 17	C2
-52	-2^2 * 13	C2
-84	-2^2 * 3 * 7	C2 x C2
-88	-2^3 * 11	C2
-91	-7 * 13	C2
-115	-5 * 23	C2
-120	-2^3 * 3 * 5	C2 x C2
-123	-3 * 41	C2
-132	-2^2 * 3 * 11	C2 x C2
-148	-2^2 * 37	C2
-168	-2^3 * 3 * 7	C2 x C2
-187	-11 * 17	C2
-195	-3 * 5 * 13	C2 x C2
-228	-2^2 * 3 * 19	C2 x C2
-232	-2^3 * 29	C2
-235	-5 * 47	C2
-267	-3 * 89	C2
-280	-2^3 * 5 * 7	C2 x C2
-312	-2^3 * 3 * 13	C2 x C2
-340	-2^2 * 5 * 17	C2 x C2
-372	-2^2 * 3 * 31	C2 x C2
-403	-13 * 31	C2
-408	-2^3 * 3 * 17	C2 x C2
-420	-2^2 * 3 * 5 * 7	C2 x C2 x C2
-427	-7 * 61	C2
-435	-3 * 5 * 29	C2 x C2
-483	-3 * 7 * 23	C2 x C2
-520	-2^3 * 5 * 13	C2 x C2
-532	-2^2 * 7 * 19	C2 x C2
-555	-3 * 5 * 37	C2 x C2
-595	-5 * 7 * 17	C2 x C2
-627	-3 * 11 * 19	C2 x C2
-660	-2^2 * 3 * 5 * 11	C2 x C2 x C2
-708	-2^2 * 3 * 59	C2 x C2
-715	-5 * 11 * 13	C2 x C2
-760	-2^3 * 5 * 19	C2 x C2
-795	-3 * 5 * 53	C2 x C2
-840	-2^3 * 3 * 5 * 7	C2 x C2 x C2
-1012	-2^2 * 11 * 23	C2 x C2
-1092	-2^2 * 3 * 7 * 13	C2 x C2 x C2
-1155	-3 * 5 * 7 * 11	C2 x C2 x C2
-1320	-2^3 * 3 * 5 * 11	C2 x C2 x C2
-1380	-2^2 * 3 * 5 * 23	C2 x C2 x C2
-1428	-2^2 * 3 * 7 * 17	C2 x C2 x C2
-1435	-5 * 7 * 41	C2 x C2
-1540	-2^2 * 5 * 7 * 11	C2 x C2 x C2
-1848	-2^3 * 3 * 7 * 11	C2 x C2 x C2
-1995	-3 * 5 * 7 * 19	C2 x C2 x C2
-3003	-3 * 7 * 11 * 13	C2 x C2 x C2
-3315	-3 * 5 * 13 * 17	C2 x C2 x C2
-5460	-2^2 * 3 * 5 * 7 * 13	C2 x C2 x C2 x C2

List of 17 fields with exponent 3

The following table is complete assuming that the generalized Riemann hypothesis is true. Without any assumption it is known that the list should be finite.

Discriminant	Factorization	Class group
-23	-23	C3
-31	-31	C3
-59	-59	C3
-83	-83	C3
-107	-107	C3
-139	-139	C3
-211	-211	C3
-283	-283	C3
-307	-307	C3
-331	-331	C3
-379	-379	C3
-499	-499	C3
-547	-547	C3
-643	-643	C3
-883	-883	C3
-907	-907	C3
-4027	-4027	C3 x C3

List of 203 fields with exponent 4

The following table is complete assuming that there are no Siegel zeros. Without any assumption there might be one missing field.

Discriminant	Factorization	Class group
-39	-3 * 13	C4
-55	-5 * 11	C4
-56	-2^3 * 7	C4
-68	-2^2 * 17	C4
-136	-2^3 * 17	C4
-155	-5 * 31	C4
-184	-2^3 * 23	C4
-203	-7 * 29	C4
-219	-3 * 73	C4
-259	-7 * 37	C4
-260	-2^2 * 5 * 13	C2 x C4
-264	-2^3 * 3 * 11	C2 x C4
-276	-2^2 * 3 * 23	C2 x C4
-291	-3 * 97	C4
-292	-2^2 * 73	C4
-308	-2^2 * 7 * 11	C2 x C4
-323	-17 * 19	C4
-328	-2^3 * 41	C4
-355	-5 * 71	C4
-388	-2^2 * 97	C4
-456	-2^3 * 3 * 19	C2 x C4
-552	-2^3 * 3 * 23	C2 x C4
-564	-2^2 * 3 * 47	C2 x C4
-568	-2^3 * 71	C4
-580	-2^2 * 5 * 29	C2 x C4
-616	-2^3 * 7 * 11	C2 x C4
-651	-3 * 7 * 31	C2 x C4
-667	-23 * 29	C4
-723	-3 * 241	C4
-763	-7 * 109	C4
-772	-2^2 * 193	C4
-820	-2^2 * 5 * 41	C2 x C4
-852	-2^2 * 3 * 71	C2 x C4
-868	-2^2 * 7 * 31	C2 x C4
-915	-3 * 5 * 61	C2 x C4
-952	-2^3 * 7 * 17	C2 x C4
-955	-5 * 191	C4
-987	-3 * 7 * 47	C2 x C4
-1003	-17 * 59	C4
-1027	-13 * 79	C4
-1032	-2^3 * 3 * 43	C2 x C4
-1060	-2^2 * 5 * 53	C2 x C4
-1128	-2^3 * 3 * 47	C2 x C4
-1131	-3 * 13 * 29	C2 x C4
-1140	-2^2 * 3 * 5 * 19	C2 x C2 x C4
-1204	-2^2 * 7 * 43	C2 x C4
-1227	-3 * 409	C4
-1240	-2^3 * 5 * 31	C2 x C4
-1243	-11 * 113	C4
-1288	-2^3 * 7 * 23	C2 x C4
-1387	-19 * 73	C4
-1411	-17 * 83	C4
-1443	-3 * 13 * 37	C2 x C4
-1507	-11 * 137	C4
-1555	-5 * 311	C4
-1560	-2^3 * 3 * 5 * 13	C2 x C2 x C4
-1635	-3 * 5 * 109	C2 x C4
-1659	-3 * 7 * 79	C2 x C4
-1672	-2^3 * 11 * 19	C2 x C4
-1716	-2^2 * 3 * 11 * 13	C2 x C2 x C4
-1752	-2^3 * 3 * 73	C2 x C4
-1768	-2^3 * 13 * 17	C2 x C4
-1771	-7 * 11 * 23	C2 x C4
-1780	-2^2 * 5 * 89	C2 x C4
-1860	-2^2 * 3 * 5 * 31	C2 x C2 x C4
-1947	-3 * 11 * 59	C2 x C4
-1992	-2^3 * 3 * 83	C2 x C4
-2020	-2^2 * 5 * 101	C2 x C4
-2035	-5 * 11 * 37	C2 x C4
-2040	-2^3 * 3 * 5 * 17	C2 x C2 x C4
-2067	-3 * 13 * 53	C2 x C4
-2139	-3 * 23 * 31	C2 x C4
-2163	-3 * 7 * 103	C2 x C4
-2212	-2^2 * 7 * 79	C2 x C4
-2244	-2^2 * 3 * 11 * 17	C2 x C2 x C4
-2280	-2^3 * 3 * 5 * 19	C2 x C2 x C4
-2379	-3 * 13 * 61	C4 x C4
-2392	-2^3 * 13 * 23	C2 x C4
-2436	-2^2 * 3 * 7 * 29	C2 x C2 x C4
-2451	-3 * 19 * 43	C2 x C4
-2580	-2^2 * 3 * 5 * 43	C2 x C2 x C4
-2632	-2^3 * 7 * 47	C2 x C4
-2667	-3 * 7 * 127	C2 x C4
-2715	-3 * 5 * 181	C2 x C4
-2755	-5 * 19 * 29	C2 x C4
-2760	-2^3 * 3 * 5 * 23	C2 x C2 x C4
-2788	-2^2 * 17 * 41	C2 x C4
-2968	-2^3 * 7 * 53	C2 x C4
-3108	-2^2 * 3 * 7 * 37	C2 x C2 x C4
-3172	-2^2 * 13 * 61	C2 x C4
-3192	-2^3 * 3 * 7 * 19	C2 x C2 x C4
-3220	-2^2 * 5 * 7 * 23	C2 x C2 x C4
-3243	-3 * 23 * 47	C2 x C4
-3355	-5 * 11 * 61	C2 x C4
-3432	-2^3 * 3 * 11 * 13	C2 x C2 x C4
-3480	-2^3 * 3 * 5 * 29	C2 x C2 x C4
-3507	-3 * 7 * 167	C2 x C4
-3588	-2^2 * 3 * 13 * 23	C2 x C2 x C4
-3640	-2^3 * 5 * 7 * 13	C2 x C2 x C4
-3795	-3 * 5 * 11 * 23	C2 x C2 x C4
-3828	-2^2 * 3 * 11 * 29	C2 x C2 x C4
-4020	-2^2 * 3 * 5 * 67	C2 x C2 x C4
-4123	-7 * 19 * 31	C2 x C4
-4180	-2^2 * 5 * 11 * 19	C2 x C2 x C4
-4260	-2^2 * 3 * 5 * 71	C2 x C2 x C4
-4323	-3 * 11 * 131	C2 x C4
-4420	-2^2 * 5 * 13 * 17	C2 x C2 x C4
-4440	-2^3 * 3 * 5 * 37	C2 x C2 x C4
-4452	-2^2 * 3 * 7 * 53	C2 x C2 x C4
-4488	-2^3 * 3 * 11 * 17	C2 x C2 x C4
-4515	-3 * 5 * 7 * 43	C2 x C2 x C4
-4740	-2^2 * 3 * 5 * 79	C2 x C2 x C4
-5083	-13 * 17 * 23	C2 x C4
-5115	-3 * 5 * 11 * 31	C2 x C2 x C4
-5160	-2^3 * 3 * 5 * 43	C2 x C2 x C4
-5187	-3 * 7 * 13 * 19	C2 x C2 x C4
-5208	-2^3 * 3 * 7 * 31	C2 x C2 x C4
-5412	-2^2 * 3 * 11 * 41	C2 x C2 x C4
-5467	-7 * 11 * 71	C2 x C4
-6052	-2^2 * 17 * 89	C4 x C4
-6123	-3 * 13 * 157	C4 x C4
-6195	-3 * 5 * 7 * 59	C2 x C2 x C4
-6307	-7 * 17 * 53	C2 x C4
-6328	-2^3 * 7 * 113	C4 x C4
-6355	-5 * 31 * 41	C4 x C4
-6360	-2^3 * 3 * 5 * 53	C2 x C4 x C4
-6420	-2^2 * 3 * 5 * 107	C2 x C2 x C4
-6580	-2^2 * 5 * 7 * 47	C2 x C2 x C4
-6612	-2^2 * 3 * 19 * 29	C2 x C2 x C4
-6708	-2^2 * 3 * 13 * 43	C2 x C2 x C4
-6820	-2^2 * 5 * 11 * 31	C2 x C2 x C4
-7035	-3 * 5 * 7 * 67	C2 x C2 x C4
-7140	-2^2 * 3 * 5 * 7 * 17	C2 x C2 x C2 x C4
-7315	-5 * 7 * 11 * 19	C2 x C2 x C4
-7395	-3 * 5 * 17 * 29	C2 x C2 x C4
-7480	-2^3 * 5 * 11 * 17	C2 x C2 x C4
-7540	-2^2 * 5 * 13 * 29	C2 x C2 x C4
-7672	-2^3 * 7 * 137	C4 x C4
-7755	-3 * 5 * 11 * 47	C2 x C2 x C4
-7995	-3 * 5 * 13 * 41	C2 x C2 x C4
-8008	-2^3 * 7 * 11 * 13	C2 x C2 x C4
-8052	-2^2 * 3 * 11 * 61	C2 x C2 x C4
-8547	-3 * 7 * 11 * 37	C2 x C2 x C4
-8580	-2^2 * 3 * 5 * 11 * 13	C2 x C2 x C2 x C4
-8680	-2^3 * 5 * 7 * 31	C2 x C2 x C4
-8715	-3 * 5 * 7 * 83	C2 x C2 x C4
-8835	-3 * 5 * 19 * 31	C2 x C2 x C4
-8932	-2^2 * 7 * 11 * 29	C2 x C2 x C4
-9240	-2^3 * 3 * 5 * 7 * 11	C2 x C2 x C2 x C4
-9595	-5 * 19 * 101	C4 x C4
-9867	-3 * 11 * 13 * 23	C2 x C2 x C4
-9955	-5 * 11 * 181	C4 x C4
-10168	-2^3 * 31 * 41	C4 x C4
-10803	-3 * 13 * 277	C4 x C4
-10920	-2^3 * 3 * 5 * 7 * 13	C2 x C2 x C2 x C4
-10948	-2^2 * 7 * 17 * 23	C2 x C2 x C4
-11067	-3 * 7 * 17 * 31	C2 x C2 x C4
-11715	-3 * 5 * 11 * 71	C2 x C2 x C4
-12180	-2^2 * 3 * 5 * 7 * 29	C2 x C2 x C2 x C4
-12595	-5 * 11 * 229	C4 x C4
-13195	-5 * 7 * 13 * 29	C2 x C2 x C4
-14008	-2^3 * 17 * 103	C4 x C4
-14155	-5 * 19 * 149	C4 x C4
-14280	-2^3 * 3 * 5 * 7 * 17	C2 x C2 x C2 x C4
-14547	-3 * 13 * 373	C4 x C4
-14763	-3 * 7 * 19 * 37	C2 x C2 x C4
-14820	-2^2 * 3 * 5 * 13 * 19	C2 x C2 x C2 x C4
-16555	-5 * 7 * 11 * 43	C2 x C2 x C4
-17220	-2^2 * 3 * 5 * 7 * 41	C2 x C2 x C2 x C4
-17427	-3 * 37 * 157	C4 x C4
-19240	-2^3 * 5 * 13 * 37	C2 x C4 x C4
-19320	-2^3 * 3 * 5 * 7 * 23	C2 x C2 x C2 x C4
-19380	-2^2 * 3 * 5 * 17 * 19	C2 x C2 x C2 x C4
-19635	-3 * 5 * 7 * 11 * 17	C2 x C2 x C2 x C4
-19947	-3 * 61 * 109	C4 x C4
-20020	-2^2 * 5 * 7 * 11 * 13	C2 x C2 x C2 x C4
-20148	-2^2 * 3 * 23 * 73	C2 x C4 x C4
-20155	-5 * 29 * 139	C4 x C4
-23640	-2^3 * 3 * 5 * 197	C2 x C4 x C4
-25608	-2^3 * 3 * 11 * 97	C2 x C4 x C4
-30340	-2^2 * 5 * 37 * 41	C2 x C4 x C4
-31395	-3 * 5 * 7 * 13 * 23	C2 x C2 x C2 x C4
-33915	-3 * 5 * 7 * 17 * 19	C2 x C2 x C2 x C4
-34840	-2^3 * 5 * 13 * 67	C2 x C4 x C4
-40755	-3 * 5 * 11 * 13 * 19	C2 x C2 x C2 x C4
-42420	-2^2 * 3 * 5 * 7 * 101	C2 x C2 x C4 x C4
-43435	-5 * 7 * 17 * 73	C2 x C4 x C4
-44115	-3 * 5 * 17 * 173	C2 x C4 x C4
-46852	-2^2 * 13 * 17 * 53	C2 x C4 x C4
-53592	-2^3 * 3 * 7 * 11 * 29	C2 x C2 x C4 x C4
-57387	-3 * 11 * 37 * 47	C2 x C4 x C4
-57715	-5 * 7 * 17 * 97	C2 x C4 x C4
-58548	-2^2 * 3 * 7 * 17 * 41	C2 x C2 x C4 x C4
-73140	-2^2 * 3 * 5 * 23 * 53	C2 x C2 x C4 x C4
-82555	-5 * 11 * 19 * 79	C2 x C4 x C4
-92568	-2^3 * 3 * 7 * 19 * 29	C2 x C2 x C4 x C4
-105315	-3 * 5 * 7 * 17 * 59	C2 x C2 x C4 x C4
-111435	-3 * 5 * 17 * 19 * 23	C2 x C2 x C4 x C4
-198660	-2^2 * 3 * 5 * 7 * 11 * 43	C2 x C2 x C2 x C4 x C4
-207480	-2^3 * 3 * 5 * 7 * 13 * 19	C2 x C2 x C2 x C4 x C4
-228228	-2^2 * 3 * 7 * 11 * 13 * 19	C2 x C2 x C2 x C4 x C4
-264180	-2^2 * 3 * 5 * 7 * 17 * 37	C2 x C2 x C2 x C4 x C4
-435435	-3 * 5 * 7 * 11 * 13 * 29	C2 x C2 x C2 x C4 x C4

List of 27 fields with exponent 5

The following table is complete assuming that the generalized Riemann hypothesis is true. Without any assumption it is known that the list should be finite.

Discriminant	Factorization	Class group
-47	-47	C5
-79	-79	C5
-103	-103	C5
-127	-127	C5
-131	-131	C5
-179	-179	C5
-227	-227	C5
-347	-347	C5
-443	-443	C5
-523	-523	C5
-571	-571	C5
-619	-619	C5
-683	-683	C5
-691	-691	C5
-739	-739	C5
-787	-787	C5
-947	-947	C5
-1051	-1051	C5
-1123	-1123	C5
-1723	-1723	C5
-1747	-1747	C5
-1867	-1867	C5
-2203	-2203	C5
-2347	-2347	C5
-2683	-2683	C5
-12451	-12451	C5 x C5
-37363	-37363	C5 x C5

List of 432 fields with exponent 6

The following table is only complete up to discriminant 3.1*10^20. It is unconditionally known that there is only a finite number of fields with exponent 6.

Discriminant	Factorization	Class group
-87	-3 * 29	C6
-104	-2^3 * 13	C6
-116	-2^2 * 29	C6
-152	-2^3 * 19	C6
-212	-2^2 * 53	C6
-231	-3 * 7 * 11	C2 x C6
-244	-2^2 * 61	C6
-247	-13 * 19	C6
-255	-3 * 5 * 17	C2 x C6
-339	-3 * 113	C6
-411	-3 * 137	C6
-424	-2^3 * 53	C6
-436	-2^2 * 109	C6
-440	-2^3 * 5 * 11	C2 x C6
-451	-11 * 41	C6
-472	-2^3 * 59	C6
-515	-5 * 103	C6
-516	-2^2 * 3 * 43	C2 x C6
-628	-2^2 * 157	C6
-680	-2^3 * 5 * 17	C2 x C6
-696	-2^3 * 3 * 29	C2 x C6
-707	-7 * 101	C6
-728	-2^3 * 7 * 13	C2 x C6
-744	-2^3 * 3 * 31	C2 x C6
-771	-3 * 257	C6
-804	-2^2 * 3 * 67	C2 x C6
-808	-2^3 * 101	C6
-835	-5 * 167	C6
-843	-3 * 281	C6
-856	-2^3 * 107	C6
-888	-2^3 * 3 * 37	C2 x C6
-948	-2^2 * 3 * 79	C2 x C6
-984	-2^3 * 3 * 41	C2 x C6
-996	-2^2 * 3 * 83	C2 x C6
-1048	-2^3 * 131	C6
-1059	-3 * 353	C6
-1099	-7 * 157	C6
-1108	-2^2 * 277	C6
-1144	-2^3 * 11 * 13	C2 x C6
-1147	-31 * 37	C6
-1192	-2^3 * 149	C6
-1203	-3 * 401	C6
-1219	-23 * 53	C6
-1235	-5 * 13 * 19	C2 x C6
-1236	-2^2 * 3 * 103	C2 x C6
-1267	-7 * 181	C6
-1272	-2^3 * 3 * 53	C2 x C6
-1315	-5 * 263	C6
-1347	-3 * 449	C6
-1363	-29 * 47	C6
-1419	-3 * 11 * 43	C2 x C6
-1432	-2^3 * 179	C6
-1464	-2^3 * 3 * 61	C2 x C6
-1480	-2^3 * 5 * 37	C2 x C6
-1491	-3 * 7 * 71	C2 x C6
-1515	-3 * 5 * 101	C2 x C6
-1547	-7 * 13 * 17	C2 x C6
-1563	-3 * 521	C6
-1572	-2^2 * 3 * 131	C2 x C6
-1588	-2^2 * 397	C6
-1603	-7 * 229	C6
-1668	-2^2 * 3 * 139	C2 x C6
-1720	-2^3 * 5 * 43	C2 x C6
-1812	-2^2 * 3 * 151	C2 x C6
-1843	-19 * 97	C6
-1892	-2^2 * 11 * 43	C2 x C6
-1915	-5 * 383	C6
-1955	-5 * 17 * 23	C2 x C6
-1963	-13 * 151	C6
-1972	-2^2 * 17 * 29	C2 x C6
-2068	-2^2 * 11 * 47	C2 x C6
-2091	-3 * 17 * 41	C2 x C6
-2132	-2^2 * 13 * 41	C2 x C6
-2148	-2^2 * 3 * 179	C2 x C6
-2184	-2^3 * 3 * 7 * 13	C2 x C2 x C6
-2227	-17 * 131	C6
-2235	-3 * 5 * 149	C2 x C6
-2260	-2^2 * 5 * 113	C2 x C6
-2283	-3 * 761	C6
-2355	-3 * 5 * 157	C2 x C6
-2387	-7 * 11 * 31	C2 x C6
-2388	-2^2 * 3 * 199	C2 x C6
-2424	-2^3 * 3 * 101	C2 x C6
-2440	-2^3 * 5 * 61	C2 x C6
-2443	-7 * 349	C6
-2472	-2^3 * 3 * 103	C2 x C6
-2515	-5 * 503	C6
-2555	-5 * 7 * 73	C2 x C6
-2563	-11 * 233	C6
-2595	-3 * 5 * 173	C2 x C6
-2635	-5 * 17 * 31	C2 x C6
-2660	-2^2 * 5 * 7 * 19	C2 x C2 x C6
-2676	-2^2 * 3 * 223	C2 x C6
-2680	-2^3 * 5 * 67	C2 x C6
-2728	-2^3 * 11 * 31	C2 x C6
-2740	-2^2 * 5 * 137	C2 x C6
-2787	-3 * 929	C6
-2795	-5 * 13 * 43	C2 x C6
-2820	-2^2 * 3 * 5 * 47	C2 x C2 x C6
-2856	-2^3 * 3 * 7 * 17	C2 x C2 x C6
-2920	-2^3 * 5 * 73	C2 x C6
-2923	-37 * 79	C6
-2955	-3 * 5 * 197	C2 x C6
-2964	-2^2 * 3 * 13 * 19	C2 x C2 x C6
-3012	-2^2 * 3 * 251	C2 x C6
-3048	-2^3 * 3 * 127	C2 x C6
-3115	-5 * 7 * 89	C2 x C6
-3235	-5 * 647	C6
-3252	-2^2 * 3 * 271	C2 x C6
-3256	-2^3 * 11 * 37	C2 x C6
-3268	-2^2 * 19 * 43	C2 x C6
-3304	-2^3 * 7 * 59	C2 x C6
-3427	-23 * 149	C6
-3444	-2^2 * 3 * 7 * 41	C2 x C2 x C6
-3451	-7 * 17 * 29	C2 x C6
-3523	-13 * 271	C6
-3540	-2^2 * 3 * 5 * 59	C2 x C2 x C6
-3619	-7 * 11 * 47	C2 x C6
-3652	-2^2 * 11 * 83	C2 x C6
-3720	-2^3 * 3 * 5 * 31	C2 x C2 x C6
-3723	-3 * 17 * 73	C2 x C6
-3763	-53 * 71	C6
-3768	-2^3 * 3 * 157	C2 x C6
-3796	-2^2 * 13 * 73	C2 x C6
-3835	-5 * 13 * 59	C2 x C6
-3864	-2^3 * 3 * 7 * 23	C2 x C2 x C6
-3876	-2^2 * 3 * 17 * 19	C2 x C2 x C6
-3880	-2^3 * 5 * 97	C2 x C6
-3892	-2^2 * 7 * 139	C2 x C6
-3955	-5 * 7 * 113	C2 x C6
-3972	-2^2 * 3 * 331	C2 x C6
-4035	-3 * 5 * 269	C2 x C6
-4120	-2^3 * 5 * 103	C2 x C6
-4147	-11 * 13 * 29	C2 x C6
-4152	-2^3 * 3 * 173	C2 x C6
-4155	-3 * 5 * 277	C2 x C6
-4360	-2^3 * 5 * 109	C2 x C6
-4587	-3 * 11 * 139	C2 x C6
-4648	-2^3 * 7 * 83	C2 x C6
-4692	-2^2 * 3 * 17 * 23	C2 x C2 x C6
-4708	-2^2 * 11 * 107	C2 x C6
-4755	-3 * 5 * 317	C2 x C6
-4795	-5 * 7 * 137	C2 x C6
-4872	-2^3 * 3 * 7 * 29	C2 x C2 x C6
-4888	-2^3 * 13 * 47	C2 x C6
-4920	-2^3 * 3 * 5 * 41	C2 x C2 x C6
-4947	-3 * 17 * 97	C2 x C6
-5016	-2^3 * 3 * 11 * 19	C2 x C2 x C6
-5032	-2^3 * 17 * 37	C2 x C6
-5035	-5 * 19 * 53	C2 x C6
-5124	-2^2 * 3 * 7 * 61	C2 x C2 x C6
-5140	-2^2 * 5 * 257	C2 x C6
-5236	-2^2 * 7 * 11 * 17	C2 x C2 x C6
-5307	-3 * 29 * 61	C2 x C6
-5320	-2^3 * 5 * 7 * 19	C2 x C2 x C6
-5395	-5 * 13 * 83	C2 x C6
-5523	-3 * 7 * 263	C2 x C6
-5595	-3 * 5 * 373	C2 x C6
-5763	-3 * 17 * 113	C2 x C6
-5811	-3 * 13 * 149	C2 x C6
-5835	-3 * 5 * 389	C2 x C6
-5928	-2^3 * 3 * 13 * 19	C2 x C2 x C6
-6072	-2^3 * 3 * 11 * 23	C2 x C2 x C6
-6132	-2^2 * 3 * 7 * 73	C2 x C2 x C6
-6180	-2^2 * 3 * 5 * 103	C2 x C2 x C6
-6216	-2^3 * 3 * 7 * 37	C2 x C2 x C6
-6232	-2^3 * 19 * 41	C2 x C6
-6235	-5 * 29 * 43	C2 x C6
-6555	-3 * 5 * 19 * 23	C2 x C2 x C6
-6603	-3 * 31 * 71	C2 x C6
-6643	-7 * 13 * 73	C2 x C6
-6699	-3 * 7 * 11 * 29	C2 x C2 x C6
-6715	-5 * 17 * 79	C2 x C6
-6888	-2^3 * 3 * 7 * 41	C2 x C2 x C6
-6916	-2^2 * 7 * 13 * 19	C2 x C2 x C6
-6955	-5 * 13 * 107	C2 x C6
-6963	-3 * 11 * 211	C2 x C6
-6987	-3 * 17 * 137	C2 x C6
-7107	-3 * 23 * 103	C2 x C6
-7320	-2^3 * 3 * 5 * 61	C2 x C2 x C6
-7332	-2^2 * 3 * 13 * 47	C2 x C2 x C6
-7620	-2^2 * 3 * 5 * 127	C2 x C2 x C6
-7683	-3 * 13 * 197	C2 x C6
-7912	-2^3 * 23 * 43	C2 x C6
-8148	-2^2 * 3 * 7 * 97	C2 x C2 x C6
-8155	-5 * 7 * 233	C2 x C6
-8211	-3 * 7 * 17 * 23	C2 x C2 x C6
-8260	-2^2 * 5 * 7 * 59	C2 x C2 x C6
-8323	-7 * 29 * 41	C2 x C6
-8395	-5 * 23 * 73	C2 x C6
-8740	-2^2 * 5 * 19 * 23	C2 x C2 x C6
-8760	-2^3 * 3 * 5 * 73	C2 x C2 x C6
-8772	-2^2 * 3 * 17 * 43	C2 x C2 x C6
-8787	-3 * 29 * 101	C2 x C6
-8827	-7 * 13 * 97	C2 x C6
-9048	-2^3 * 3 * 13 * 29	C2 x C2 x C6
-9139	-13 * 19 * 37	C2 x C6
-9384	-2^3 * 3 * 17 * 23	C2 x C2 x C6
-9480	-2^3 * 3 * 5 * 79	C2 x C2 x C6
-9492	-2^2 * 3 * 7 * 113	C2 x C2 x C6
-9672	-2^3 * 3 * 13 * 31	C2 x C2 x C6
-9748	-2^2 * 2437	C3 x C6
-9843	-3 * 17 * 193	C2 x C6
-9940	-2^2 * 5 * 7 * 71	C2 x C2 x C6
-10120	-2^3 * 5 * 11 * 23	C2 x C2 x C6
-10212	-2^2 * 3 * 23 * 37	C2 x C2 x C6
-10248	-2^3 * 3 * 7 * 61	C2 x C2 x C6
-10360	-2^3 * 5 * 7 * 37	C2 x C2 x C6
-10452	-2^2 * 3 * 13 * 67	C2 x C2 x C6
-10488	-2^3 * 3 * 19 * 23	C2 x C2 x C6
-10707	-3 * 43 * 83	C2 x C6
-10740	-2^2 * 3 * 5 * 179	C2 x C2 x C6
-10788	-2^2 * 3 * 29 * 31	C2 x C2 x C6
-10795	-5 * 17 * 127	C2 x C6
-10915	-5 * 37 * 59	C2 x C6
-11155	-5 * 23 * 97	C2 x C6
-11220	-2^2 * 3 * 5 * 11 * 17	C2 x C2 x C2 x C6
-11235	-3 * 5 * 7 * 107	C2 x C2 x C6
-11620	-2^2 * 5 * 7 * 83	C2 x C2 x C6
-11748	-2^2 * 3 * 11 * 89	C2 x C2 x C6
-11803	-11 * 29 * 37	C2 x C6
-11928	-2^3 * 3 * 7 * 71	C2 x C2 x C6
-12067	-11 * 1097	C3 x C6
-12243	-3 * 7 * 11 * 53	C2 x C2 x C6
-12376	-2^3 * 7 * 13 * 17	C2 x C2 x C6
-12408	-2^3 * 3 * 11 * 47	C2 x C2 x C6
-12628	-2^2 * 7 * 11 * 41	C2 x C2 x C6
-12760	-2^3 * 5 * 11 * 29	C2 x C2 x C6
-13035	-3 * 5 * 11 * 79	C2 x C2 x C6
-13080	-2^3 * 3 * 5 * 109	C2 x C2 x C6
-13395	-3 * 5 * 19 * 47	C2 x C2 x C6
-13668	-2^2 * 3 * 17 * 67	C2 x C2 x C6
-13780	-2^2 * 5 * 13 * 53	C2 x C2 x C6
-14212	-2^2 * 11 * 17 * 19	C2 x C2 x C6
-14235	-3 * 5 * 13 * 73	C2 x C2 x C6
-14260	-2^2 * 5 * 23 * 31	C2 x C2 x C6
-14443	-11 * 13 * 101	C2 x C6
-14532	-2^2 * 3 * 7 * 173	C2 x C2 x C6
-14595	-3 * 5 * 7 * 139	C2 x C2 x C6
-14835	-3 * 5 * 23 * 43	C2 x C2 x C6
-14952	-2^3 * 3 * 7 * 89	C2 x C2 x C6
-14980	-2^2 * 5 * 7 * 107	C2 x C2 x C6
-15283	-17 * 29 * 31	C2 x C6
-15540	-2^2 * 3 * 5 * 7 * 37	C2 x C2 x C2 x C6
-15544	-2^3 * 29 * 67	C6 x C6
-15555	-3 * 5 * 17 * 61	C2 x C2 x C6
-15640	-2^3 * 5 * 17 * 23	C2 x C2 x C6
-15652	-2^2 * 7 * 13 * 43	C2 x C2 x C6
-15960	-2^3 * 3 * 5 * 7 * 19	C2 x C2 x C2 x C6
-16107	-3 * 7 * 13 * 59	C2 x C2 x C6
-16627	-13 * 1279	C3 x C6
-16872	-2^3 * 3 * 19 * 37	C2 x C2 x C6
-17043	-3 * 13 * 19 * 23	C2 x C2 x C6
-17131	-37 * 463	C3 x C6
-17160	-2^3 * 3 * 5 * 11 * 13	C2 x C2 x C2 x C6
-17556	-2^2 * 3 * 7 * 11 * 19	C2 x C2 x C2 x C6
-17940	-2^2 * 3 * 5 * 13 * 23	C2 x C2 x C2 x C6
-18040	-2^3 * 5 * 11 * 41	C2 x C2 x C6
-18088	-2^3 * 7 * 17 * 19	C2 x C2 x C6
-18340	-2^2 * 5 * 7 * 131	C2 x C2 x C6
-18555	-3 * 5 * 1237	C6 x C6
-18795	-3 * 5 * 7 * 179	C2 x C2 x C6
-18915	-3 * 5 * 13 * 97	C2 x C2 x C6
-19140	-2^2 * 3 * 5 * 11 * 29	C2 x C2 x C2 x C6
-19651	-43 * 457	C3 x C6
-19803	-3 * 7 * 23 * 41	C2 x C2 x C6
-20355	-3 * 5 * 23 * 59	C2 x C2 x C6
-20568	-2^3 * 3 * 857	C6 x C6
-20955	-3 * 5 * 11 * 127	C2 x C2 x C6
-20995	-5 * 13 * 17 * 19	C2 x C2 x C6
-21112	-2^3 * 7 * 13 * 29	C2 x C2 x C6
-21252	-2^2 * 3 * 7 * 11 * 23	C2 x C2 x C2 x C6
-22260	-2^2 * 3 * 5 * 7 * 53	C2 x C2 x C2 x C6
-22395	-3 * 5 * 1493	C6 x C6
-22440	-2^3 * 3 * 5 * 11 * 17	C2 x C2 x C2 x C6
-22443	-3 * 7481	C3 x C6
-23115	-3 * 5 * 23 * 67	C2 x C2 x C6
-23188	-2^2 * 11 * 17 * 31	C2 x C2 x C6
-23683	-11 * 2153	C3 x C6
-24115	-5 * 7 * 13 * 53	C2 x C2 x C6
-24123	-3 * 11 * 17 * 43	C2 x C2 x C6
-24180	-2^2 * 3 * 5 * 13 * 31	C2 x C2 x C2 x C6
-24340	-2^2 * 5 * 1217	C6 x C6
-24360	-2^3 * 3 * 5 * 7 * 29	C2 x C2 x C2 x C6
-24388	-2^2 * 7 * 13 * 67	C2 x C2 x C6
-24420	-2^2 * 3 * 5 * 11 * 37	C2 x C2 x C2 x C6
-24915	-3 * 5 * 11 * 151	C2 x C2 x C6
-24955	-5 * 7 * 23 * 31	C2 x C2 x C6
-25347	-3 * 7 * 17 * 71	C2 x C2 x C6
-25707	-3 * 11 * 19 * 41	C2 x C2 x C6
-25755	-3 * 5 * 17 * 101	C2 x C2 x C6
-25795	-5 * 7 * 11 * 67	C2 x C2 x C6
-26040	-2^3 * 3 * 5 * 7 * 31	C2 x C2 x C2 x C6
-26187	-3 * 7 * 29 * 43	C2 x C2 x C6
-26520	-2^3 * 3 * 5 * 13 * 17	C2 x C2 x C2 x C6
-26760	-2^3 * 3 * 5 * 223	C2 x C6 x C6
-27060	-2^2 * 3 * 5 * 11 * 41	C2 x C2 x C2 x C6
-27115	-5 * 11 * 17 * 29	C2 x C2 x C6
-27156	-2^2 * 3 * 31 * 73	C2 x C6 x C6
-27435	-3 * 5 * 31 * 59	C2 x C2 x C6
-27640	-2^3 * 5 * 691	C6 x C6
-28644	-2^2 * 3 * 7 * 11 * 31	C2 x C2 x C2 x C6
-29172	-2^2 * 3 * 11 * 13 * 17	C2 x C2 x C2 x C6
-29640	-2^3 * 3 * 5 * 13 * 19	C2 x C2 x C2 x C6
-30360	-2^3 * 3 * 5 * 11 * 23	C2 x C2 x C2 x C6
-30660	-2^2 * 3 * 5 * 7 * 73	C2 x C2 x C2 x C6
-31620	-2^2 * 3 * 5 * 17 * 31	C2 x C2 x C2 x C6
-31908	-2^2 * 3 * 2659	C6 x C6
-32968	-2^3 * 13 * 317	C6 x C6
-33060	-2^2 * 3 * 5 * 19 * 29	C2 x C2 x C2 x C6
-34027	-7 * 4861	C3 x C6
-34507	-11 * 3137	C3 x C6
-34827	-3 * 13 * 19 * 47	C2 x C2 x C6
-34867	-7 * 17 * 293	C6 x C6
-34980	-2^2 * 3 * 5 * 11 * 53	C2 x C2 x C2 x C6
-35112	-2^3 * 3 * 7 * 11 * 19	C2 x C2 x C2 x C6
-36120	-2^3 * 3 * 5 * 7 * 43	C2 x C2 x C2 x C6
-36708	-2^2 * 3 * 7 * 19 * 23	C2 x C2 x C2 x C6
-37128	-2^3 * 3 * 7 * 13 * 17	C2 x C2 x C2 x C6
-37219	-7 * 13 * 409	C6 x C6
-37555	-5 * 7 * 29 * 37	C2 x C2 x C6
-38280	-2^3 * 3 * 5 * 11 * 29	C2 x C2 x C2 x C6
-39480	-2^3 * 3 * 5 * 7 * 47	C2 x C2 x C2 x C6
-39732	-2^2 * 3 * 7 * 11 * 43	C2 x C2 x C2 x C6
-40299	-3 * 7 * 19 * 101	C2 x C6 x C6
-40692	-2^2 * 3 * 3391	C6 x C6
-41412	-2^2 * 3 * 7 * 17 * 29	C2 x C2 x C2 x C6
-41860	-2^2 * 5 * 7 * 13 * 23	C2 x C2 x C2 x C6
-42315	-3 * 5 * 7 * 13 * 31	C2 x C2 x C2 x C6
-42427	-7 * 11 * 19 * 29	C2 x C2 x C6
-42619	-17 * 23 * 109	C6 x C6
-43428	-2^2 * 3 * 7 * 11 * 47	C2 x C2 x C2 x C6
-43827	-3 * 7 * 2087	C6 x C6
-44004	-2^2 * 3 * 19 * 193	C2 x C6 x C6
-45220	-2^2 * 5 * 7 * 17 * 19	C2 x C2 x C2 x C6
-45835	-5 * 89 * 103	C6 x C6
-46587	-3 * 53 * 293	C6 x C6
-46740	-2^2 * 3 * 5 * 19 * 41	C2 x C2 x C2 x C6
-47355	-3 * 5 * 7 * 11 * 41	C2 x C2 x C2 x C6
-48052	-2^2 * 41 * 293	C6 x C6
-48472	-2^3 * 73 * 83	C6 x C6
-49128	-2^3 * 3 * 23 * 89	C2 x C6 x C6
-49812	-2^2 * 3 * 7 * 593	C2 x C6 x C6
-50388	-2^2 * 3 * 13 * 17 * 19	C2 x C2 x C2 x C6
-51051	-3 * 7 * 11 * 13 * 17	C2 x C2 x C2 x C6
-51348	-2^2 * 3 * 11 * 389	C2 x C6 x C6
-52360	-2^3 * 5 * 7 * 11 * 17	C2 x C2 x C2 x C6
-54195	-3 * 5 * 3613	C6 x C6
-58920	-2^3 * 3 * 5 * 491	C2 x C6 x C6
-60099	-3 * 13 * 23 * 67	C2 x C6 x C6
-63492	-2^2 * 3 * 11 * 13 * 37	C2 x C2 x C2 x C6
-64155	-3 * 5 * 7 * 13 * 47	C2 x C2 x C2 x C6
-67480	-2^3 * 5 * 7 * 241	C2 x C6 x C6
-70035	-3 * 5 * 7 * 23 * 29	C2 x C2 x C2 x C6
-72435	-3 * 5 * 11 * 439	C2 x C6 x C6
-78180	-2^2 * 3 * 5 * 1303	C2 x C6 x C6
-78708	-2^2 * 3 * 7 * 937	C2 x C6 x C6
-81867	-3 * 29 * 941	C6 x C6
-83395	-5 * 13 * 1283	C6 x C6
-84072	-2^3 * 3 * 31 * 113	C2 x C6 x C6
-86955	-3 * 5 * 11 * 17 * 31	C2 x C2 x C2 x C6
-87720	-2^3 * 3 * 5 * 17 * 43	C2 x C2 x C6 x C6
-87780	-2^2 * 3 * 5 * 7 * 11 * 19	C2 x C2 x C2 x C2 x C6
-92827	-7 * 89 * 149	C6 x C6
-94395	-3 * 5 * 7 * 29 * 31	C2 x C2 x C2 x C6
-95448	-2^3 * 3 * 41 * 97	C2 x C6 x C6
-100488	-2^3 * 3 * 53 * 79	C2 x C6 x C6
-106260	-2^2 * 3 * 5 * 7 * 11 * 23	C2 x C2 x C2 x C2 x C6
-107848	-2^3 * 13 * 17 * 61	C2 x C6 x C6
-112795	-5 * 17 * 1327	C6 x C6
-115780	-2^2 * 5 * 7 * 827	C2 x C6 x C6
-116083	-11 * 61 * 173	C6 x C6
-117480	-2^3 * 3 * 5 * 11 * 89	C2 x C2 x C6 x C6
-119112	-2^3 * 3 * 7 * 709	C2 x C6 x C6
-121720	-2^3 * 5 * 17 * 179	C2 x C6 x C6
-125652	-2^2 * 3 * 37 * 283	C2 x C6 x C6
-132328	-2^3 * 7 * 17 * 139	C2 x C6 x C6
-137067	-3 * 7 * 61 * 107	C2 x C6 x C6
-138468	-2^2 * 3 * 11 * 1049	C2 x C6 x C6
-145860	-2^2 * 3 * 5 * 11 * 13 * 17	C2 x C2 x C2 x C2 x C6
-148852	-2^2 * 11 * 17 * 199	C2 x C6 x C6
-152355	-3 * 5 * 7 * 1451	C2 x C6 x C6
-154212	-2^2 * 3 * 71 * 181	C2 x C6 x C6
-180840	-2^3 * 3 * 5 * 11 * 137	C2 x C2 x C6 x C6
-183012	-2^2 * 3 * 101 * 151	C2 x C6 x C6
-183768	-2^3 * 3 * 13 * 19 * 31	C2 x C2 x C6 x C6
-192003	-3 * 7 * 41 * 223	C2 x C6 x C6
-195960	-2^3 * 3 * 5 * 23 * 71	C2 x C2 x C6 x C6
-198795	-3 * 5 * 29 * 457	C2 x C6 x C6
-199348	-2^2 * 19 * 43 * 61	C2 x C6 x C6
-204568	-2^3 * 7 * 13 * 281	C2 x C6 x C6
-211432	-2^3 * 13 * 19 * 107	C2 x C6 x C6
-214008	-2^3 * 3 * 37 * 241	C2 x C6 x C6
-224580	-2^2 * 3 * 5 * 19 * 197	C2 x C2 x C6 x C6
-231240	-2^3 * 3 * 5 * 41 * 47	C2 x C2 x C6 x C6
-287155	-5 * 11 * 23 * 227	C2 x C6 x C6
-303160	-2^3 * 5 * 11 * 13 * 53	C2 x C2 x C6 x C6
-315723	-3 * 19 * 29 * 191	C2 x C6 x C6
-341715	-3 * 5 * 11 * 19 * 109	C2 x C2 x C6 x C6
-343380	-2^2 * 3 * 5 * 59 * 97	C2 x C2 x C6 x C6
-352968	-2^3 * 3 * 7 * 11 * 191	C2 x C2 x C6 x C6
-393108	-2^2 * 3 * 17 * 41 * 47	C2 x C2 x C6 x C6
-394420	-2^2 * 5 * 13 * 37 * 41	C2 x C2 x C6 x C6
-397155	-3 * 5 * 11 * 29 * 83	C2 x C2 x C6 x C6
-404547	-3 * 11 * 13 * 23 * 41	C2 x C2 x C6 x C6
-419640	-2^3 * 3 * 5 * 13 * 269	C2 x C2 x C6 x C6
-423640	-2^3 * 5 * 7 * 17 * 89	C2 x C2 x C6 x C6
-453435	-3 * 5 * 19 * 37 * 43	C2 x C2 x C6 x C6
-458920	-2^3 * 5 * 7 * 11 * 149	C2 x C2 x C6 x C6
-507892	-2^2 * 7 * 11 * 17 * 97	C2 x C2 x C6 x C6
-512715	-3 * 5 * 7 * 19 * 257	C2 x C2 x C6 x C6
-522795	-3 * 5 * 7 * 13 * 383	C2 x C2 x C6 x C6
-603460	-2^2 * 5 * 11 * 13 * 211	C2 x C2 x C6 x C6
-668980	-2^2 * 5 * 13 * 31 * 83	C2 x C2 x C6 x C6
-680043	-3 * 7 * 13 * 47 * 53	C2 x C2 x C6 x C6
-697620	-2^2 * 3 * 5 * 7 * 11 * 151	C2 x C2 x C2 x C6 x C6
-740355	-3 * 5 * 7 * 11 * 641	C2 x C2 x C6 x C6
-740532	-2^2 * 3 * 13 * 47 * 101	C2 x C2 x C6 x C6
-742980	-2^2 * 3 * 5 * 7 * 29 * 61	C2 x C2 x C2 x C6 x C6
-820120	-2^3 * 5 * 7 * 29 * 101	C2 x C2 x C6 x C6
-899283	-3 * 7 * 11 * 17 * 229	C2 x C2 x C6 x C6
-941640	-2^3 * 3 * 5 * 7 * 19 * 59	C2 x C2 x C2 x C6 x C6
-1162392	-2^3 * 3 * 7 * 11 * 17 * 37	C2 x C2 x C2 x C6 x C6
-1172395	-5 * 7 * 19 * 41 * 43	C2 x C2 x C6 x C6
-1185240	-2^3 * 3 * 5 * 7 * 17 * 83	C2 x C2 x C2 x C6 x C6
-1196052	-2^2 * 3 * 11 * 13 * 17 * 41	C2 x C2 x C2 x C6 x C6
-1199220	-2^2 * 3 * 5 * 11 * 23 * 79	C2 x C2 x C2 x C6 x C6
-1282260	-2^2 * 3 * 5 * 7 * 43 * 71	C2 x C2 x C2 x C6 x C6
-1702155	-3 * 5 * 7 * 13 * 29 * 43	C2 x C2 x C2 x C6 x C6
-3892980	-2^2 * 3 * 5 * 7 * 13 * 23 * 31	C2 x C2 x C2 x C2 x C6 x C6
-4696692	-2^2 * 3 * 7 * 11 * 13 * 17 * 23	C2 x C2 x C2 x C2 x C6 x C6
-5761140	-2^2 * 3 * 5 * 7 * 11 * 29 * 43	C2 x C2 x C2 x C2 x C6 x C6

List of 33 fields with exponent 7

The following table is only complete up to discriminant 3.1*10^20.

Discriminant	Factorization	Class group
-71	-71	C7
-151	-151	C7
-223	-223	C7
-251	-251	C7
-463	-463	C7
-467	-467	C7
-487	-487	C7
-587	-587	C7
-811	-811	C7
-827	-827	C7
-859	-859	C7
-1163	-1163	C7
-1171	-1171	C7
-1483	-1483	C7
-1523	-1523	C7
-1627	-1627	C7
-1787	-1787	C7
-1987	-1987	C7
-2011	-2011	C7
-2083	-2083	C7
-2179	-2179	C7
-2251	-2251	C7
-2467	-2467	C7
-2707	-2707	C7
-3019	-3019	C7
-3067	-3067	C7
-3187	-3187	C7
-3907	-3907	C7
-4603	-4603	C7
-5107	-5107	C7
-5923	-5923	C7
-63499	-63499	C7 x C7
-118843	-118843	C7 x C7

List of 778 fields with exponent 8

The following table is complete assuming that there are no Siegel zeros. Without any assumption there might be one missing field.

Discriminant	Factorization	Class group
-95	-5 * 19	C8
-111	-3 * 37	C8
-164	-2^2 * 41	C8
-183	-3 * 61	C8
-248	-2^3 * 31	C8
-295	-5 * 59	C8
-299	-13 * 23	C8
-371	-7 * 53	C8
-376	-2^3 * 47	C8
-395	-5 * 79	C8
-399	-3 * 7 * 19	C2 x C8
-452	-2^2 * 113	C8
-548	-2^2 * 137	C8
-579	-3 * 193	C8
-583	-11 * 53	C8
-632	-2^3 * 79	C8
-644	-2^2 * 7 * 23	C2 x C8
-663	-3 * 13 * 17	C2 x C8
-712	-2^3 * 89	C8
-740	-2^2 * 5 * 37	C2 x C8
-884	-2^2 * 13 * 17	C2 x C8
-903	-3 * 7 * 43	C2 x C8
-904	-2^3 * 113	C8
-939	-3 * 313	C8
-979	-11 * 89	C8
-995	-5 * 199	C8
-1015	-5 * 7 * 29	C2 x C8
-1023	-3 * 11 * 31	C2 x C8
-1043	-7 * 149	C8
-1195	-5 * 239	C8
-1220	-2^2 * 5 * 61	C2 x C8
-1252	-2^2 * 313	C8
-1299	-3 * 433	C8
-1339	-13 * 103	C8
-1348	-2^2 * 337	C8
-1416	-2^3 * 3 * 59	C2 x C8
-1508	-2^2 * 13 * 29	C2 x C8
-1528	-2^3 * 191	C8
-1595	-5 * 11 * 29	C2 x C8
-1608	-2^3 * 3 * 67	C2 x C8
-1624	-2^3 * 7 * 29	C2 x C8
-1640	-2^3 * 5 * 41	C2 x C8
-1651	-13 * 127	C8
-1731	-3 * 577	C8
-1795	-5 * 359	C8
-1803	-3 * 601	C8
-1828	-2^2 * 457	C8
-1864	-2^3 * 233	C8
-1876	-2^2 * 7 * 67	C2 x C8
-1912	-2^3 * 239	C8
-1924	-2^2 * 13 * 37	C2 x C8
-1939	-7 * 277	C8
-2004	-2^2 * 3 * 167	C2 x C8
-2059	-29 * 71	C8
-2072	-2^3 * 7 * 37	C2 x C8
-2211	-3 * 11 * 67	C2 x C8
-2248	-2^3 * 281	C8
-2292	-2^2 * 3 * 191	C2 x C8
-2296	-2^3 * 7 * 41	C2 x C8
-2307	-3 * 769	C8
-2308	-2^2 * 577	C8
-2323	-23 * 101	C8
-2328	-2^3 * 3 * 97	C2 x C8
-2356	-2^2 * 19 * 31	C2 x C8
-2395	-5 * 479	C8
-2419	-41 * 59	C8
-2568	-2^3 * 3 * 107	C2 x C8
-2584	-2^3 * 17 * 19	C2 x C8
-2587	-13 * 199	C8
-2611	-7 * 373	C8
-2739	-3 * 11 * 83	C2 x C8
-2827	-11 * 257	C8
-2868	-2^2 * 3 * 239	C2 x C8
-2884	-2^2 * 7 * 103	C2 x C8
-2947	-7 * 421	C8
-2980	-2^2 * 5 * 149	C2 x C8
-2995	-5 * 599	C8
-3080	-2^3 * 5 * 7 * 11	C2 x C2 x C8
-3140	-2^2 * 5 * 157	C2 x C8
-3144	-2^3 * 3 * 131	C2 x C8
-3160	-2^3 * 5 * 79	C2 x C8
-3171	-3 * 7 * 151	C2 x C8
-3336	-2^3 * 3 * 139	C2 x C8
-3363	-3 * 19 * 59	C2 x C8
-3403	-41 * 83	C8
-3435	-3 * 5 * 229	C2 x C8
-3448	-2^3 * 431	C8
-3460	-2^2 * 5 * 173	C2 x C8
-3531	-3 * 11 * 107	C2 x C8
-3556	-2^2 * 7 * 127	C2 x C8
-3595	-5 * 719	C8
-3732	-2^2 * 3 * 311	C2 x C8
-3752	-2^3 * 7 * 67	C2 x C8
-3784	-2^3 * 11 * 43	C2 x C8
-3787	-7 * 541	C8
-3819	-3 * 19 * 67	C2 x C8
-3883	-11 * 353	C8
-3939	-3 * 13 * 101	C2 x C8
-3963	-3 * 1321	C8
-3976	-2^3 * 7 * 71	C2 x C8
-4008	-2^3 * 3 * 167	C2 x C8
-4179	-3 * 7 * 199	C2 x C8
-4195	-5 * 839	C8
-4216	-2^3 * 17 * 31	C2 x C8
-4228	-2^2 * 7 * 151	C2 x C8
-4251	-3 * 13 * 109	C2 x C8
-4267	-17 * 251	C8
-4324	-2^2 * 23 * 47	C2 x C8
-4340	-2^2 * 5 * 7 * 31	C2 x C2 x C8
-4387	-41 * 107	C8
-4596	-2^2 * 3 * 383	C2 x C8
-4683	-3 * 7 * 223	C2 x C8
-4712	-2^3 * 19 * 31	C2 x C8
-4747	-47 * 101	C8
-4843	-29 * 167	C8
-4867	-31 * 157	C8
-4884	-2^2 * 3 * 11 * 37	C2 x C2 x C8
-4899	-3 * 23 * 71	C2 x C8
-4980	-2^2 * 3 * 5 * 83	C2 x C2 x C8
-4984	-2^3 * 7 * 89	C2 x C8
-5380	-2^2 * 5 * 269	C2 x C8
-5428	-2^2 * 23 * 59	C2 x C8
-5572	-2^2 * 7 * 199	C2 x C8
-5587	-37 * 151	C8
-5640	-2^3 * 3 * 5 * 47	C2 x C2 x C8
-5668	-2^2 * 13 * 109	C2 x C8
-5707	-13 * 439	C8
-5720	-2^3 * 5 * 11 * 13	C2 x C2 x C8
-5795	-5 * 19 * 61	C4 x C8
-5848	-2^3 * 17 * 43	C2 x C8
-5860	-2^2 * 5 * 293	C2 x C8
-5883	-3 * 37 * 53	C2 x C8
-5896	-2^3 * 11 * 67	C2 x C8
-5907	-3 * 11 * 179	C2 x C8
-5908	-2^2 * 7 * 211	C2 x C8
-5947	-19 * 313	C8
-5992	-2^3 * 7 * 107	C2 x C8
-5995	-5 * 11 * 109	C2 x C8
-6040	-2^3 * 5 * 151	C2 x C8
-6099	-3 * 19 * 107	C2 x C8
-6148	-2^2 * 29 * 53	C2 x C8
-6312	-2^3 * 3 * 263	C2 x C8
-6315	-3 * 5 * 421	C2 x C8
-6392	-2^3 * 17 * 47	C4 x C8
-6440	-2^3 * 5 * 7 * 23	C2 x C2 x C8
-6532	-2^2 * 23 * 71	C2 x C8
-6747	-3 * 13 * 173	C2 x C8
-6771	-3 * 37 * 61	C2 x C8
-6792	-2^3 * 3 * 283	C2 x C8
-6868	-2^2 * 17 * 101	C2 x C8
-6923	-7 * 23 * 43	C2 x C8
-6952	-2^3 * 11 * 79	C2 x C8
-7059	-3 * 13 * 181	C4 x C8
-7176	-2^3 * 3 * 13 * 23	C2 x C2 x C8
-7347	-3 * 31 * 79	C2 x C8
-7368	-2^3 * 3 * 307	C2 x C8
-7491	-3 * 11 * 227	C2 x C8
-7579	-11 * 13 * 53	C2 x C8
-7588	-2^2 * 7 * 271	C2 x C8
-7707	-3 * 7 * 367	C2 x C8
-7752	-2^3 * 3 * 17 * 19	C2 x C2 x C8
-7780	-2^2 * 5 * 389	C2 x C8
-7828	-2^2 * 19 * 103	C2 x C8
-7843	-11 * 23 * 31	C2 x C8
-7896	-2^3 * 3 * 7 * 47	C2 x C2 x C8
-7923	-3 * 19 * 139	C2 x C8
-8040	-2^3 * 3 * 5 * 67	C2 x C2 x C8
-8043	-3 * 7 * 383	C2 x C8
-8184	-2^3 * 3 * 11 * 31	C2 x C2 x C8
-8283	-3 * 11 * 251	C2 x C8
-8308	-2^2 * 31 * 67	C2 x C8
-8340	-2^2 * 3 * 5 * 139	C2 x C2 x C8
-8515	-5 * 13 * 131	C2 x C8
-8520	-2^3 * 3 * 5 * 71	C2 x C2 x C8
-8555	-5 * 29 * 59	C4 x C8
-8635	-5 * 11 * 157	C2 x C8
-8643	-3 * 43 * 67	C2 x C8
-8968	-2^3 * 19 * 59	C2 x C8
-9060	-2^2 * 3 * 5 * 151	C2 x C2 x C8
-9156	-2^2 * 3 * 7 * 109	C2 x C2 x C8
-9219	-3 * 7 * 439	C2 x C8
-9316	-2^2 * 17 * 137	C4 x C8
-9348	-2^2 * 3 * 19 * 41	C2 x C2 x C8
-9412	-2^2 * 13 * 181	C2 x C8
-9435	-3 * 5 * 17 * 37	C2 x C2 x C8
-9460	-2^2 * 5 * 11 * 43	C2 x C2 x C8
-9483	-3 * 29 * 109	C2 x C8
-9588	-2^2 * 3 * 17 * 47	C2 x C2 x C8
-9640	-2^3 * 5 * 241	C2 x C8
-9768	-2^3 * 3 * 11 * 37	C2 x C2 x C8
-9780	-2^2 * 3 * 5 * 163	C2 x C2 x C8
-9835	-5 * 7 * 281	C2 x C8
-9860	-2^2 * 5 * 17 * 29	C2 x C2 x C8
-9880	-2^3 * 5 * 13 * 19	C2 x C2 x C8
-9912	-2^3 * 3 * 7 * 59	C2 x C2 x C8
-9960	-2^3 * 3 * 5 * 83	C2 x C2 x C8
-10132	-2^2 * 17 * 149	C2 x C8
-10203	-3 * 19 * 179	C2 x C8
-10227	-3 * 7 * 487	C2 x C8
-10387	-13 * 17 * 47	C2 x C8
-10420	-2^2 * 5 * 521	C2 x C8
-10563	-3 * 7 * 503	C2 x C8
-10635	-3 * 5 * 709	C2 x C8
-10660	-2^2 * 5 * 13 * 41	C2 x C2 x C8
-10824	-2^3 * 3 * 11 * 41	C2 x C2 x C8
-10868	-2^2 * 11 * 13 * 19	C2 x C2 x C8
-11092	-2^2 * 47 * 59	C2 x C8
-11284	-2^2 * 7 * 13 * 31	C2 x C2 x C8
-11316	-2^2 * 3 * 23 * 41	C2 x C2 x C8
-11460	-2^2 * 3 * 5 * 191	C2 x C2 x C8
-11523	-3 * 23 * 167	C2 x C8
-11571	-3 * 7 * 19 * 29	C2 x C2 x C8
-11572	-2^2 * 11 * 263	C2 x C8
-11635	-5 * 13 * 179	C2 x C8
-11739	-3 * 7 * 13 * 43	C2 x C2 x C8
-11832	-2^3 * 3 * 17 * 29	C2 x C2 x C8
-11940	-2^2 * 3 * 5 * 199	C2 x C2 x C8
-12027	-3 * 19 * 211	C2 x C8
-12040	-2^3 * 5 * 7 * 43	C2 x C2 x C8
-12084	-2^2 * 3 * 19 * 53	C2 x C2 x C8
-12259	-13 * 23 * 41	C2 x C8
-12580	-2^2 * 5 * 17 * 37	C2 x C2 x C8
-12660	-2^2 * 3 * 5 * 211	C2 x C2 x C8
-12747	-3 * 7 * 607	C2 x C8
-12772	-2^2 * 31 * 103	C2 x C8
-12792	-2^3 * 3 * 13 * 41	C2 x C2 x C8
-12804	-2^2 * 3 * 11 * 97	C2 x C2 x C8
-12835	-5 * 17 * 151	C2 x C8
-12840	-2^3 * 3 * 5 * 107	C2 x C2 x C8
-12859	-7 * 11 * 167	C2 x C8
-12948	-2^2 * 3 * 13 * 83	C2 x C2 x C8
-13048	-2^3 * 7 * 233	C4 x C8
-13192	-2^3 * 17 * 97	C2 x C8
-13272	-2^3 * 3 * 7 * 79	C2 x C2 x C8
-13288	-2^3 * 11 * 151	C2 x C8
-13332	-2^2 * 3 * 11 * 101	C2 x C2 x C8
-13363	-7 * 23 * 83	C2 x C8
-13432	-2^3 * 23 * 73	C4 x C8
-13515	-3 * 5 * 17 * 53	C2 x C2 x C8
-13560	-2^3 * 3 * 5 * 113	C2 x C2 x C8
-13755	-3 * 5 * 7 * 131	C2 x C2 x C8
-13795	-5 * 31 * 89	C2 x C8
-13827	-3 * 11 * 419	C2 x C8
-13908	-2^2 * 3 * 19 * 61	C2 x C2 x C8
-14388	-2^2 * 3 * 11 * 109	C2 x C2 x C8
-14707	-7 * 11 * 191	C2 x C8
-14795	-5 * 11 * 269	C4 x C8
-15067	-13 * 19 * 61	C2 x C8
-15387	-3 * 23 * 223	C2 x C8
-15715	-5 * 7 * 449	C2 x C8
-15736	-2^3 * 7 * 281	C4 x C8
-16008	-2^3 * 3 * 23 * 29	C2 x C2 x C8
-16027	-11 * 31 * 47	C2 x C8
-16195	-5 * 41 * 79	C2 x C8
-16212	-2^2 * 3 * 7 * 193	C2 x C2 x C8
-16440	-2^3 * 3 * 5 * 137	C2 x C2 x C8
-16548	-2^2 * 3 * 7 * 197	C2 x C2 x C8
-16692	-2^2 * 3 * 13 * 107	C2 x C2 x C8
-16779	-3 * 7 * 17 * 47	C2 x C2 x C8
-16835	-5 * 7 * 13 * 37	C2 x C2 x C8
-16995	-3 * 5 * 11 * 103	C2 x C2 x C8
-17080	-2^3 * 5 * 7 * 61	C2 x C2 x C8
-17112	-2^3 * 3 * 23 * 31	C2 x C2 x C8
-17115	-3 * 5 * 7 * 163	C2 x C2 x C8
-17227	-7 * 23 * 107	C2 x C8
-17272	-2^3 * 17 * 127	C4 x C8
-17347	-11 * 19 * 83	C2 x C8
-17355	-3 * 5 * 13 * 89	C2 x C2 x C8
-17515	-5 * 31 * 113	C2 x C8
-17688	-2^3 * 3 * 11 * 67	C2 x C2 x C8
-17880	-2^3 * 3 * 5 * 149	C2 x C2 x C8
-18291	-3 * 7 * 13 * 67	C2 x C2 x C8
-18403	-7 * 11 * 239	C2 x C8
-18408	-2^3 * 3 * 13 * 59	C2 x C2 x C8
-18532	-2^2 * 41 * 113	C4 x C8
-18564	-2^2 * 3 * 7 * 13 * 17	C2 x C2 x C2 x C8
-18676	-2^2 * 7 * 23 * 29	C2 x C2 x C8
-18715	-5 * 19 * 197	C2 x C8
-18760	-2^3 * 5 * 7 * 67	C2 x C2 x C8
-18907	-7 * 37 * 73	C2 x C8
-19108	-2^2 * 17 * 281	C4 x C8
-19195	-5 * 11 * 349	C2 x C8
-19227	-3 * 13 * 17 * 29	C2 x C2 x C8
-19272	-2^3 * 3 * 11 * 73	C2 x C2 x C8
-19560	-2^3 * 3 * 5 * 163	C2 x C2 x C8
-19608	-2^3 * 3 * 19 * 43	C2 x C2 x C8
-19720	-2^3 * 5 * 17 * 29	C2 x C2 x C8
-19812	-2^2 * 3 * 13 * 127	C2 x C2 x C8
-19987	-11 * 23 * 79	C2 x C8
-19995	-3 * 5 * 31 * 43	C2 x C2 x C8
-20091	-3 * 37 * 181	C4 x C8
-20163	-3 * 11 * 13 * 47	C2 x C2 x C8
-20235	-3 * 5 * 19 * 71	C2 x C2 x C8
-20740	-2^2 * 5 * 17 * 61	C2 x C2 x C8
-20760	-2^3 * 3 * 5 * 173	C2 x C4 x C8
-20868	-2^2 * 3 * 37 * 47	C2 x C2 x C8
-21147	-3 * 7 * 19 * 53	C2 x C2 x C8
-21243	-3 * 73 * 97	C4 x C8
-21571	-11 * 37 * 53	C4 x C8
-21715	-5 * 43 * 101	C2 x C8
-21736	-2^3 * 11 * 13 * 19	C2 x C2 x C8
-21828	-2^2 * 3 * 17 * 107	C2 x C2 x C8
-21835	-5 * 11 * 397	C2 x C8
-21912	-2^3 * 3 * 11 * 83	C2 x C2 x C8
-22152	-2^3 * 3 * 13 * 71	C2 x C2 x C8
-22155	-3 * 5 * 7 * 211	C2 x C2 x C8
-22243	-13 * 29 * 59	C2 x C8
-22360	-2^3 * 5 * 13 * 43	C2 x C2 x C8
-22372	-2^2 * 7 * 17 * 47	C2 x C2 x C8
-22420	-2^2 * 5 * 19 * 59	C2 x C2 x C8
-22456	-2^3 * 7 * 401	C4 x C8
-22515	-3 * 5 * 19 * 79	C2 x C2 x C8
-22632	-2^3 * 3 * 23 * 41	C2 x C2 x C8
-22740	-2^2 * 3 * 5 * 379	C2 x C2 x C8
-22763	-13 * 17 * 103	C4 x C8
-22792	-2^3 * 7 * 11 * 37	C2 x C2 x C8
-22971	-3 * 13 * 19 * 31	C2 x C2 x C8
-23028	-2^2 * 3 * 19 * 101	C2 x C2 x C8
-23140	-2^2 * 5 * 13 * 89	C2 x C2 x C8
-23155	-5 * 11 * 421	C4 x C8
-23268	-2^2 * 3 * 7 * 277	C2 x C2 x C8
-23380	-2^2 * 5 * 7 * 167	C2 x C2 x C8
-23460	-2^2 * 3 * 5 * 17 * 23	C2 x C2 x C2 x C8
-23835	-3 * 5 * 7 * 227	C2 x C2 x C8
-23892	-2^2 * 3 * 11 * 181	C2 x C2 x C8
-23944	-2^3 * 41 * 73	C4 x C8
-24004	-2^2 * 17 * 353	C4 x C8
-24024	-2^3 * 3 * 7 * 11 * 13	C2 x C2 x C2 x C8
-24035	-5 * 11 * 19 * 23	C2 x C2 x C8
-24168	-2^3 * 3 * 19 * 53	C2 x C2 x C8
-24472	-2^3 * 7 * 19 * 23	C2 x C2 x C8
-25080	-2^3 * 3 * 5 * 11 * 19	C2 x C2 x C2 x C8
-25419	-3 * 37 * 229	C4 x C8
-25620	-2^2 * 3 * 5 * 7 * 61	C2 x C2 x C2 x C8
-25960	-2^3 * 5 * 11 * 59	C2 x C2 x C8
-25988	-2^2 * 73 * 89	C8 x C8
-26180	-2^2 * 5 * 7 * 11 * 17	C2 x C2 x C2 x C8
-26488	-2^3 * 7 * 11 * 43	C2 x C2 x C8
-26772	-2^2 * 3 * 23 * 97	C2 x C2 x C8
-26980	-2^2 * 5 * 19 * 71	C2 x C2 x C8
-27307	-7 * 47 * 83	C2 x C8
-27412	-2^2 * 7 * 11 * 89	C2 x C2 x C8
-28083	-3 * 11 * 23 * 37	C2 x C2 x C8
-28120	-2^3 * 5 * 19 * 37	C2 x C2 x C8
-28203	-3 * 7 * 17 * 79	C2 x C2 x C8
-28308	-2^2 * 3 * 7 * 337	C2 x C2 x C8
-28840	-2^3 * 5 * 7 * 103	C2 x C2 x C8
-29380	-2^2 * 5 * 13 * 113	C2 x C2 x C8
-29512	-2^3 * 7 * 17 * 31	C2 x C2 x C8
-29523	-3 * 13 * 757	C4 x C8
-29667	-3 * 11 * 29 * 31	C2 x C2 x C8
-29892	-2^2 * 3 * 47 * 53	C2 x C2 x C8
-30328	-2^3 * 17 * 223	C4 x C8
-30552	-2^3 * 3 * 19 * 67	C2 x C2 x C8
-30580	-2^2 * 5 * 11 * 139	C2 x C2 x C8
-31080	-2^3 * 3 * 5 * 7 * 37	C2 x C2 x C2 x C8
-31515	-3 * 5 * 11 * 191	C2 x C2 x C8
-31668	-2^2 * 3 * 7 * 13 * 29	C2 x C2 x C2 x C8
-31864	-2^3 * 7 * 569	C8 x C8
-32235	-3 * 5 * 7 * 307	C2 x C2 x C8
-32331	-3 * 13 * 829	C8 x C8
-32395	-5 * 11 * 19 * 31	C2 x C2 x C8
-33288	-2^3 * 3 * 19 * 73	C2 x C4 x C8
-33540	-2^2 * 3 * 5 * 13 * 43	C2 x C2 x C2 x C8
-33592	-2^3 * 13 * 17 * 19	C2 x C2 x C8
-33672	-2^3 * 3 * 23 * 61	C2 x C2 x C8
-33748	-2^2 * 11 * 13 * 59	C2 x C2 x C8
-34552	-2^3 * 7 * 617	C4 x C8
-34755	-3 * 5 * 7 * 331	C2 x C2 x C8
-34804	-2^2 * 7 * 11 * 113	C2 x C4 x C8
-35155	-5 * 79 * 89	C4 x C8
-35160	-2^3 * 3 * 5 * 293	C2 x C4 x C8
-35380	-2^2 * 5 * 29 * 61	C2 x C2 x C8
-35512	-2^3 * 23 * 193	C4 x C8
-35763	-3 * 7 * 13 * 131	C2 x C2 x C8
-36355	-5 * 11 * 661	C4 x C8
-36363	-3 * 17 * 23 * 31	C2 x C2 x C8
-36472	-2^3 * 47 * 97	C4 x C8
-36660	-2^2 * 3 * 5 * 13 * 47	C2 x C2 x C2 x C8
-36915	-3 * 5 * 23 * 107	C2 x C2 x C8
-36955	-5 * 19 * 389	C4 x C8
-37380	-2^2 * 3 * 5 * 7 * 89	C2 x C2 x C2 x C8
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