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 3.1·1020. 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.
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 |
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 |
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 |
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 |
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 |
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-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 |
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 |
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 |
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-366760 | -2^3 * 5 * 53 * 173 | C2 x C8 x C8 |
-369835 | -5 * 17 * 19 * 229 | C2 x C4 x C8 |
-374088 | -2^3 * 3 * 11 * 13 * 109 | C2 x C2 x C4 x C8 |
-374595 | -3 * 5 * 13 * 17 * 113 | C2 x C2 x C4 x C8 |
-383380 | -2^2 * 5 * 29 * 661 | C2 x C8 x C8 |
-385795 | -5 * 19 * 31 * 131 | C2 x C4 x C8 |
-386628 | -2^2 * 3 * 11 * 29 * 101 | C2 x C2 x C4 x C8 |
-409960 | -2^3 * 5 * 37 * 277 | C2 x C8 x C8 |
-417435 | -3 * 5 * 17 * 1637 | C2 x C8 x C8 |
-428835 | -3 * 5 * 11 * 23 * 113 | C2 x C2 x C4 x C8 |
-433540 | -2^2 * 5 * 53 * 409 | C2 x C8 x C8 |
-443667 | -3 * 7 * 37 * 571 | C2 x C8 x C8 |
-448035 | -3 * 5 * 7 * 17 * 251 | C2 x C2 x C4 x C8 |
-457620 | -2^2 * 3 * 5 * 29 * 263 | C2 x C2 x C4 x C8 |
-474628 | -2^2 * 7 * 11 * 23 * 67 | C2 x C2 x C4 x C8 |
-480907 | -7 * 23 * 29 * 103 | C2 x C4 x C8 |
-488523 | -3 * 7 * 43 * 541 | C2 x C8 x C8 |
-489787 | -17 * 47 * 613 | C8 x C8 |
-492388 | -2^2 * 13 * 17 * 557 | C2 x C8 x C8 |
-502660 | -2^2 * 5 * 41 * 613 | C2 x C8 x C8 |
-503659 | -13 * 17 * 43 * 53 | C4 x C4 x C8 |
-505960 | -2^3 * 5 * 7 * 13 * 139 | C2 x C2 x C4 x C8 |
-514228 | -2^2 * 11 * 13 * 29 * 31 | C2 x C2 x C4 x C8 |
-516243 | -3 * 7 * 13 * 31 * 61 | C2 x C2 x C4 x C8 |
-518635 | -5 * 13 * 79 * 101 | C2 x C8 x C8 |
-519915 | -3 * 5 * 11 * 23 * 137 | C2 x C2 x C4 x C8 |
-549915 | -3 * 5 * 61 * 601 | C2 x C8 x C8 |
-553540 | -2^2 * 5 * 13 * 2129 | C2 x C8 x C8 |
-557832 | -2^3 * 3 * 11 * 2113 | C2 x C8 x C8 |
-559083 | -3 * 7 * 79 * 337 | C2 x C8 x C8 |
-563640 | -2^3 * 3 * 5 * 7 * 11 * 61 | C2 x C2 x C2 x C4 x C8 |
-563955 | -3 * 5 * 7 * 41 * 131 | C2 x C2 x C4 x C8 |
-566040 | -2^3 * 3 * 5 * 53 * 89 | C2 x C2 x C8 x C8 |
-568888 | -2^3 * 17 * 47 * 89 | C4 x C4 x C8 |
-585915 | -3 * 5 * 11 * 53 * 67 | C2 x C2 x C4 x C8 |
-590835 | -3 * 5 * 7 * 17 * 331 | C2 x C2 x C4 x C8 |
-603988 | -2^2 * 7 * 11 * 37 * 53 | C2 x C2 x C4 x C8 |
-611403 | -3 * 13 * 61 * 257 | C2 x C8 x C8 |
-630147 | -3 * 7 * 37 * 811 | C2 x C8 x C8 |
-644280 | -2^3 * 3 * 5 * 7 * 13 * 59 | C2 x C2 x C2 x C4 x C8 |
-646195 | -5 * 11 * 31 * 379 | C2 x C8 x C8 |
-674520 | -2^3 * 3 * 5 * 7 * 11 * 73 | C2 x C2 x C2 x C4 x C8 |
-688755 | -3 * 5 * 17 * 37 * 73 | C2 x C2 x C4 x C8 |
-692692 | -2^2 * 7 * 11 * 13 * 173 | C2 x C2 x C4 x C8 |
-703560 | -2^3 * 3 * 5 * 11 * 13 * 41 | C2 x C2 x C2 x C4 x C8 |
-705432 | -2^3 * 3 * 7 * 13 * 17 * 19 | C2 x C2 x C2 x C4 x C8 |
-713883 | -3 * 47 * 61 * 83 | C2 x C8 x C8 |
-743028 | -2^2 * 3 * 11 * 13 * 433 | C2 x C2 x C8 x C8 |
-788307 | -3 * 13 * 17 * 29 * 41 | C2 x C2 x C4 x C8 |
-806520 | -2^3 * 3 * 5 * 11 * 13 * 47 | C2 x C2 x C2 x C4 x C8 |
-809115 | -3 * 5 * 17 * 19 * 167 | C2 x C2 x C8 x C8 |
-834360 | -2^3 * 3 * 5 * 17 * 409 | C2 x C2 x C8 x C8 |
-844440 | -2^3 * 3 * 5 * 31 * 227 | C2 x C2 x C8 x C8 |
-850795 | -5 * 11 * 31 * 499 | C2 x C8 x C8 |
-857220 | -2^2 * 3 * 5 * 7 * 13 * 157 | C2 x C2 x C2 x C4 x C8 |
-876315 | -3 * 5 * 11 * 47 * 113 | C2 x C2 x C8 x C8 |
-876435 | -3 * 5 * 7 * 17 * 491 | C2 x C2 x C8 x C8 |
-894660 | -2^2 * 3 * 5 * 13 * 31 * 37 | C2 x C2 x C2 x C4 x C8 |
-937560 | -2^3 * 3 * 5 * 13 * 601 | C2 x C2 x C8 x C8 |
-959803 | -13 * 17 * 43 * 101 | C4 x C4 x C8 |
-963235 | -5 * 7 * 13 * 29 * 73 | C2 x C2 x C4 x C8 |
-980148 | -2^2 * 3 * 13 * 61 * 103 | C2 x C2 x C8 x C8 |
-990964 | -2^2 * 13 * 17 * 19 * 59 | C2 x C2 x C8 x C8 |
-1039795 | -5 * 29 * 71 * 101 | C2 x C8 x C8 |
-1065540 | -2^2 * 3 * 5 * 7 * 43 * 59 | C2 x C2 x C2 x C4 x C8 |
-1076712 | -2^3 * 3 * 7 * 13 * 17 * 29 | C2 x C2 x C2 x C4 x C8 |
-1115268 | -2^2 * 3 * 7 * 11 * 17 * 71 | C2 x C2 x C2 x C4 x C8 |
-1232340 | -2^2 * 3 * 5 * 19 * 23 * 47 | C2 x C2 x C2 x C4 x C8 |
-1278552 | -2^3 * 3 * 11 * 29 * 167 | C2 x C2 x C8 x C8 |
-1291620 | -2^2 * 3 * 5 * 11 * 19 * 103 | C2 x C2 x C2 x C4 x C8 |
-1307523 | -3 * 7 * 19 * 29 * 113 | C2 x C2 x C8 x C8 |
-1362760 | -2^3 * 5 * 7 * 31 * 157 | C2 x C2 x C8 x C8 |
-1364808 | -2^3 * 3 * 19 * 41 * 73 | C2 x C2 x C8 x C8 |
-1368840 | -2^3 * 3 * 5 * 11 * 17 * 61 | C2 x C2 x C2 x C4 x C8 |
-1390180 | -2^2 * 5 * 11 * 71 * 89 | C2 x C2 x C8 x C8 |
-1408740 | -2^2 * 3 * 5 * 53 * 443 | C2 x C2 x C8 x C8 |
-1446907 | -7 * 11 * 19 * 23 * 43 | C2 x C2 x C4 x C8 |
-1472115 | -3 * 5 * 17 * 23 * 251 | C2 x C2 x C8 x C8 |
-1490952 | -2^3 * 3 * 23 * 37 * 73 | C2 x C2 x C8 x C8 |
-1519960 | -2^3 * 5 * 13 * 37 * 79 | C2 x C2 x C8 x C8 |
-1562484 | -2^2 * 3 * 7 * 11 * 19 * 89 | C2 x C2 x C2 x C8 x C8 |
-1567720 | -2^3 * 5 * 7 * 11 * 509 | C2 x C2 x C8 x C8 |
-1592115 | -3 * 5 * 7 * 59 * 257 | C2 x C2 x C8 x C8 |
-1707288 | -2^3 * 3 * 11 * 29 * 223 | C2 x C2 x C8 x C8 |
-1842715 | -5 * 7 * 17 * 19 * 163 | C2 x C2 x C8 x C8 |
-1928595 | -3 * 5 * 19 * 67 * 101 | C2 x C2 x C8 x C8 |
-1935507 | -3 * 7 * 37 * 47 * 53 | C2 x C2 x C8 x C8 |
-1937635 | -5 * 7 * 23 * 29 * 83 | C2 x C2 x C8 x C8 |
-2043195 | -3 * 5 * 7 * 11 * 29 * 61 | C2 x C2 x C2 x C4 x C8 |
-2045323 | -7 * 37 * 53 * 149 | C4 x C4 x C8 |
-2078440 | -2^3 * 5 * 7 * 13 * 571 | C2 x C2 x C8 x C8 |
-2095060 | -2^2 * 5 * 11 * 89 * 107 | C2 x C2 x C8 x C8 |
-2103220 | -2^2 * 5 * 7 * 83 * 181 | C2 x C2 x C8 x C8 |
-2104707 | -3 * 11 * 23 * 47 * 59 | C2 x C2 x C8 x C8 |
-2188920 | -2^3 * 3 * 5 * 17 * 29 * 37 | C2 x C2 x C4 x C4 x C8 |
-2209467 | -3 * 13 * 181 * 313 | C4 x C8 x C8 |
-2229123 | -3 * 13 * 61 * 937 | C4 x C8 x C8 |
-2309307 | -3 * 7 * 11 * 13 * 769 | C2 x C2 x C8 x C8 |
-2345595 | -3 * 5 * 7 * 89 * 251 | C2 x C2 x C8 x C8 |
-2376660 | -2^2 * 3 * 5 * 11 * 13 * 277 | C2 x C2 x C2 x C8 x C8 |
-2401867 | -13 * 23 * 29 * 277 | C4 x C8 x C8 |
-2420635 | -5 * 7 * 23 * 31 * 97 | C2 x C2 x C8 x C8 |
-2443155 | -3 * 5 * 11 * 13 * 17 * 67 | C2 x C2 x C2 x C4 x C8 |
-2523928 | -2^3 * 11 * 23 * 29 * 43 | C2 x C2 x C8 x C8 |
-2999560 | -2^3 * 5 * 31 * 41 * 59 | C2 x C2 x C8 x C8 |
-3040888 | -2^3 * 41 * 73 * 127 | C4 x C8 x C8 |
-3081540 | -2^2 * 3 * 5 * 7 * 11 * 23 * 29 | C2 x C2 x C2 x C2 x C4 x C8 |
-3121035 | -3 * 5 * 19 * 47 * 233 | C2 x C2 x C8 x C8 |
-3352020 | -2^2 * 3 * 5 * 7 * 23 * 347 | C2 x C2 x C2 x C8 x C8 |
-3452020 | -2^2 * 5 * 11 * 13 * 17 * 71 | C2 x C2 x C2 x C8 x C8 |
-3468595 | -5 * 13 * 17 * 43 * 73 | C2 x C2 x C8 x C8 |
-3536260 | -2^2 * 5 * 7 * 13 * 29 * 67 | C2 x C2 x C2 x C8 x C8 |
-3598980 | -2^2 * 3 * 5 * 7 * 11 * 19 * 41 | C2 x C2 x C2 x C2 x C4 x C8 |
-3608760 | -2^3 * 3 * 5 * 17 * 29 * 61 | C2 x C2 x C2 x C8 x C8 |
-4800772 | -2^2 * 41 * 73 * 401 | C4 x C8 x C8 |
-4886995 | -5 * 31 * 41 * 769 | C4 x C8 x C8 |
-4903395 | -3 * 5 * 7 * 17 * 41 * 67 | C2 x C2 x C2 x C8 x C8 |
-5358340 | -2^2 * 5 * 13 * 37 * 557 | C2 x C4 x C8 x C8 |
-5692440 | -2^3 * 3 * 5 * 13 * 41 * 89 | C2 x C2 x C4 x C8 x C8 |
-6005748 | -2^2 * 3 * 7 * 19 * 53 * 71 | C2 x C2 x C2 x C8 x C8 |
-6174168 | -2^3 * 3 * 7 * 11 * 13 * 257 | C2 x C2 x C2 x C8 x C8 |
-6585987 | -3 * 17 * 29 * 61 * 73 | C2 x C2 x C8 x C8 |
-7538388 | -2^2 * 3 * 11 * 13 * 23 * 191 | C2 x C2 x C2 x C8 x C8 |
-11148180 | -2^2 * 3 * 5 * 29 * 43 * 149 | C2 x C2 x C4 x C8 x C8 |
-12517428 | -2^2 * 3 * 7 * 11 * 19 * 23 * 31 | C2 x C2 x C2 x C4 x C4 x C8 |
-15337315 | -5 * 7 * 17 * 149 * 173 | C2 x C4 x C8 x C8 |
-15898740 | -2^2 * 3 * 5 * 11 * 13 * 17 * 109 | C2 x C2 x C2 x C2 x C8 x C8 |
-17168515 | -5 * 7 * 13 * 97 * 389 | C2 x C4 x C8 x C8 |
-28663635 | -3 * 5 * 7 * 11 * 13 * 23 * 83 | C2 x C2 x C2 x C2 x C8 x C8 |
-29493555 | -3 * 5 * 7 * 13 * 17 * 31 * 41 | C2 x C2 x C2 x C2 x C8 x C8 |
-31078723 | -13 * 43 * 53 * 1049 | C8 x C8 x C8 |
-430950520 | -2^3 * 5 * 7 * 11 * 13 * 47 * 229 | C2 x C2 x C2 x C8 x C8 x C8 |
C-Programs
The following two c-programs can be used to compute all imaginary quadratic number fields with exponent less or equal to 8 and discriminant bound 3.1·1020.
Julia Programs
The following file contains the Julia code. You need to install Julia including the Hecke and the Markdown package. The function M1 computes the imginary quadratic number fields of given exponent. The functions M2a, M2b, M3a, and M3b compute the corresponding families.
Download File