The database contains 19,869,443 number fields.

Distribution by signature

There is one row for each degree $n$, and the column is indexed by $r_2$, the number of non-real embeddings, so the signature is $(n-2r_2, r_2)$. Percentages for individual entries are taken with the degree. Percentages for the column All give the proportion of fields of a given degree among all fields in the database.
$n \backslash r_2$ All
1 1 1
100.00% 0.00%
2 684,218 684,578 1,368,796
49.99% 50.01% 6.89%
3 205,160 663,362 868,522
23.62% 76.38% 4.37%
4 703,770 674,910 1,578,163 2,956,843
23.80% 22.83% 53.37% 14.88%
5 322,135 316,464 684,948 1,323,547
24.34% 23.91% 51.75% 6.66%
6 852,564 436,953 2,086,940 2,372,302 5,748,759
14.83% 7.60% 36.30% 41.27% 28.93%
7 946 23,975 436,253 702,543 1,163,717
0.08% 2.06% 37.49% 60.37% 5.86%
8 109,722 330,522 608,027 540,615 1,496,646 3,085,532
3.56% 10.71% 19.71% 17.52% 48.51% 15.53%
9 8,485 2,959 8,932 92,923 121,792 235,091
3.61% 1.26% 3.80% 39.53% 51.81% 1.18%
10 262,148 9,083 79,530 451,556 1,124,403 820,856 2,747,576
9.54% 0.33% 2.89% 16.43% 40.92% 29.88% 13.83%
11 54 6 8 19 636 873 1,596
3.38% 0.38% 0.50% 1.19% 39.85% 54.70% 0.01%
12 3,569 846 4,604 10,091 37,595 51,939 111,729 220,373
1.62% 0.38% 2.09% 4.58% 17.06% 23.57% 50.70% 1.11%
13 28 2 2 2 10 51 180 275
10.18% 0.73% 0.73% 0.73% 3.64% 18.55% 65.45% 0.00%
14 267 18 53 47 122 39 259 758 1,563
17.08% 1.15% 3.39% 3.01% 7.81% 2.50% 16.57% 48.50% 0.01%
15 282 13 68 38 88 82 221 247 1,039
27.14% 1.25% 6.54% 3.66% 8.47% 7.89% 21.27% 23.77% 0.01%
16 5,825 115 2,685 511 12,511 1,451 12,608 2,247 34,459 72,412
8.04% 0.16% 3.71% 0.71% 17.28% 2.00% 17.41% 3.10% 47.59% 0.36%
17 17 2 2 1 2 2 7 36 149 218
7.80% 0.92% 0.92% 0.46% 0.92% 0.92% 3.21% 16.51% 68.35% 0.00%
18 2,834 90 783 588 1,449 498 2,720 754 1,658 5,728 17,102
16.57% 0.53% 4.58% 3.44% 8.47% 2.91% 15.90% 4.41% 9.69% 33.49% 0.09%
19 14 2 2 2 1 2 2 2 31 139 197
7.11% 1.02% 1.02% 1.02% 0.51% 1.02% 1.02% 1.02% 15.74% 70.56% 0.00%
20 2,198 99 1,004 260 2,730 949 3,073 847 4,248 1,023 10,457 26,888
8.17% 0.37% 3.73% 0.97% 10.15% 3.53% 11.43% 3.15% 15.80% 3.80% 38.89% 0.14%
21 216 19 36 49 48 26 96 80 54 149 191 964
22.41% 1.97% 3.73% 5.08% 4.98% 2.70% 9.96% 8.30% 5.60% 15.46% 19.81% 0.00%
22 128 15 38 27 110 69 140 50 99 31 188 235 1,130
11.33% 1.33% 3.36% 2.39% 9.73% 6.11% 12.39% 4.42% 8.76% 2.74% 16.64% 20.80% 0.01%
23 14 0 2 2 2 2 2 2 2 2 32 134 196
7.14% 0.00% 1.02% 1.02% 1.02% 1.02% 1.02% 1.02% 1.02% 1.02% 16.33% 68.37% 0.00%
Total 3,164,595 3,144,033 5,492,080 4,171,576 2,798,145 876,839 131,037 5,023 40,700 7,072 10,868 369

Distribution by Galois group

For each degree $3\leq n\leq 7$, we give the breakdown of fields in the database by Galois group. Columns are indexed by $t$, the $t$-number. Degrees $1$ and $2$ are omitted because there is only one possible Galois group for each. Percentages show the proportions are with respect to the number of fields in the database of that degree.
$n \backslash t$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 All
3 4,029 864,493 868,522
0.46% 99.54%
4 21,692 117,034 1,991,604 29,534 796,979 2,956,843
0.73% 3.96% 67.36% 1.00% 26.95%
5 1,003 16,194 28,682 114,826 1,162,842 1,323,547
0.08% 1.22% 2.17% 8.68% 87.86%
6 15,959 24,494 237,228 5,261 14,622 178,674 244,008 46,108 64,259 2,319 676,991 110,104 1,946,131 1,011,145 9,721 1,161,735 5,748,759
0.28% 0.43% 4.13% 0.09% 0.25% 3.11% 4.24% 0.80% 1.12% 0.04% 11.78% 1.92% 33.85% 17.59% 0.17% 20.21%
7 129 2,101 86 1,423 40,860 56,886 1,062,232 1,163,717
0.01% 0.18% 0.01% 0.12% 3.51% 4.89% 91.28%

Four families of groups have a representative which appears as a Galois group in every degree $n$, $C_n$, the cyclic group of order $n$, $D_n$, the dihedral group of order $2n$, $A_n$, the alternating group on $n$ letters, and $S_n$, the symmetric group on $n$ letters. Percentages are based on the number of fields in the database of the degree.
$n$ 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
$C_n$ $=A_3$ 21,692 1,003 15,959 129 6,206 284 1,802 28 2,097 14 309 96 552 11 1,143 8 773 70 166 9
0.73% 0.08% 0.28% 0.01% 0.20% 0.12% 0.07% 1.75% 0.95% 5.09% 19.77% 9.24% 0.76% 5.05% 6.68% 4.06% 2.87% 7.26% 14.69% 4.59%
$D_n$ $=S_3$ 1,991,604 16,194 237,228 2,101 30,358 706 7,317 57 158 41 33 53 86 12 10 11 9 3 8 10
67.36% 1.22% 4.13% 0.18% 0.98% 0.30% 0.27% 3.57% 0.07% 14.91% 2.11% 5.10% 0.12% 5.50% 0.06% 5.58% 0.03% 0.31% 0.71% 5.10%
$A_n$ 4,029 29,534 114,826 9,721 56,886 125 636 1,210 79 9 5 5 5 6 3 2 1 1 2 1 2
0.46% 1.00% 8.68% 0.17% 4.89% 0.00% 0.27% 0.04% 4.95% 0.00% 1.82% 0.32% 0.48% 0.01% 1.38% 0.01% 0.51% 0.00% 0.21% 0.09% 1.02%
$S_n$ 864,493 796,979 1,162,842 1,161,735 1,062,232 4,254 3,394 5,148 1,143 162 194 160 173 171 173 165 164 165 168 168 165
99.54% 26.95% 87.86% 20.21% 91.28% 0.14% 1.44% 0.19% 71.62% 0.07% 70.55% 10.24% 16.65% 0.24% 79.36% 0.96% 83.25% 0.61% 17.43% 14.87% 84.18%

Distribution by class number

Class numbers have been computed for 99.84% of the fields in the database. Some of these class numbers have been computed under the assumption of the Generalized Riemann Hypothesis.

For main table entries, the percentage is out of fields of that degree for which we have computed the class number. For row totals, the percentage is the number of fields of that degree for which we have computed the class number divided by the total number of fields of that degree. For column totals, percentages are based on the total number of fields for which we have computed the class number. Finally, the lower right corner gives the percentage of fields in the database for which we have computed the class number.
$n$ $h=1$ $1<h\leq 10$ $10<h\leq 10^2$ $10^{2}<h\leq 10^{3}$ $10^{3}<h\leq 10^{4}$ $10^{4}<h\leq 10^{5}$ $10^{5}<h\leq 10^{6}$ $10^{6}<h\leq 10^{7}$ $10^{7}<h\leq 10^{8}$ $10^{8}<h\leq 10^{9}$ $10^{9}<h\leq 10^{10}$ $10^{10}<h\leq 10^{11}$ Total
$2$ 177,168 424,904 123,439 584,815 58,470 1,368,796
12.94% 31.04% 9.02% 42.72% 4.27% 100.00%
$3$ 478,943 324,146 56,215 7,027 1,885 280 24 868,520
55.14% 37.32% 6.47% 0.81% 0.22% 0.03% 0.00% 100.00%
$4$ 1,028,892 812,998 375,923 389,529 265,611 80,826 2,935 90 1 2,956,805
34.80% 27.50% 12.71% 13.17% 8.98% 2.73% 0.10% 0.00% 0.00% 100.00%
$5$ 1,161,094 153,943 8,014 476 19 1 1,323,547
87.73% 11.63% 0.61% 0.04% 0.00% 0.00% 100.00%
$6$ 2,837,375 2,052,831 643,882 161,017 51,706 1,934 8 5,748,753
49.36% 35.71% 11.20% 2.80% 0.90% 0.03% 0.00% 100.00%
$7$ 1,137,252 26,241 218 6 1,163,717
97.73% 2.25% 0.02% 0.00% 100.00%
$8$ 1,172,632 1,081,831 567,600 186,098 49,597 14,513 5,559 3,225 1,212 306 76 8 3,082,662
38.04% 35.09% 18.41% 6.04% 1.61% 0.47% 0.18% 0.10% 0.04% 0.01% 0.00% 0.00% 99.91%
$9$ 168,996 61,490 4,477 128 235,091
71.89% 26.16% 1.90% 0.05% 100.00%
$10$ 1,745,201 838,341 123,953 38,131 1,081 564 292 4 2,747,567
63.52% 30.51% 4.51% 1.39% 0.04% 0.02% 0.01% 0.00% 100.00%
$11$ 1,482 87 25 1,594
92.97% 5.46% 1.57% 99.87%
$12$ 65,707 104,746 38,679 6,515 1,879 1,622 939 240 17 220,344
29.82% 47.54% 17.55% 2.96% 0.85% 0.74% 0.43% 0.11% 0.01% 99.99%
$13$ 264 11 275
96.00% 4.00% 100.00%
$14$ 834 491 56 24 39 41 37 26 8 1 1,557
53.56% 31.54% 3.60% 1.54% 2.50% 2.63% 2.38% 1.67% 0.51% 0.06% 99.62%
$15$ 873 147 8 2 1,030
84.76% 14.27% 0.78% 0.19% 99.13%
$16$ 26,325 22,082 7,888 4,555 3,856 2,677 2,085 1,382 707 307 98 24 72,019
36.55% 30.66% 10.95% 6.32% 5.35% 3.72% 2.90% 1.92% 0.98% 0.43% 0.14% 0.03% 99.46%
$17$ 199 12 6 217
91.71% 5.53% 2.76% 99.54%
$18$ 8,920 4,529 1,222 831 337 226 221 218 152 110 41 3 16,811
53.06% 26.94% 7.27% 4.94% 2.00% 1.34% 1.31% 1.30% 0.90% 0.65% 0.24% 0.02% 98.30%
$19$ 185 12 197
93.91% 6.09% 100.00%
$20$ 14,894 6,715 2,138 675 545 385 363 278 227 127 56 18 26,433
56.35% 25.40% 8.09% 2.55% 2.06% 1.46% 1.37% 1.05% 0.86% 0.48% 0.21% 0.07% 98.31%
$21$ 776 168 3 947
81.94% 17.74% 0.32% 98.24%
$22$ 863 117 17 4 6 12 20 13 9 2 1 2 1,066
80.96% 10.98% 1.59% 0.38% 0.56% 1.13% 1.88% 1.22% 0.84% 0.19% 0.09% 0.19% 94.34%
$23$ 181 13 1 195
92.82% 6.67% 0.51% 99.49%
Total 10,030,295 5,917,235 1,954,379 1,380,268 435,716 104,119 13,675 6,636 3,288 1,472 570 157 19,838,144
50.56% 29.83% 9.85% 6.96% 2.20% 0.52% 0.07% 0.03% 0.02% 0.01% 0.00% 0.00% 99.84%

Distribution of class number one fields by signature

A number field has class number one if and only if its ring of integers is a unique factorization domain.
The table below has one row for each degree $n$, and the columns are indexed by $r_2$, as above. Percentages for individual entries are with respect to all fields in the database with that signature for which we have computed the class number. An entry is not shown if we have not computed the class number for any fields with that combination of degree and signature.
$n \backslash r_2$
1 1
100.00%
2 177,159 9
25.89% 0.00%
3 156,575 322,368
76.32% 48.60%
4 452,363 364,005 212,524
64.28% 53.93% 13.47%
5 300,892 297,191 563,011
93.41% 93.91% 82.20%
6 464,535 318,016 847,759 1,207,065
54.49% 72.78% 40.62% 50.88%
7 889 23,968 425,305 687,090
93.97% 99.97% 97.49% 97.80%
8 75,380 272,805 282,737 203,799 337,911
68.70% 82.54% 46.50% 37.81% 22.60%
9 5,480 2,762 8,148 57,148 95,458
64.58% 93.34% 91.22% 61.50% 78.38%
10 199,196 5,262 41,438 292,460 717,497 489,348
75.99% 57.93% 52.10% 64.77% 63.81% 59.61%
11 42 6 8 19 609 798
80.77% 100.00% 100.00% 100.00% 95.75% 91.41%
12 1,268 608 3,072 6,204 18,880 21,849 13,826
35.53% 71.87% 66.72% 61.48% 50.22% 42.07% 12.38%
13 28 2 2 2 10 50 170
100.00% 100.00% 100.00% 100.00% 100.00% 98.04% 94.44%
14 189 18 48 47 109 37 225 161
71.59% 100.00% 96.00% 100.00% 89.34% 94.87% 86.87% 21.24%
15 205 13 62 36 76 74 202 205
75.09% 100.00% 91.18% 94.74% 86.36% 90.24% 91.40% 83.00%
16 3,145 109 2,071 461 7,174 988 5,640 1,022 5,715
54.17% 94.78% 77.19% 90.22% 57.57% 68.14% 44.77% 45.50% 16.74%
17 14 2 2 1 2 2 7 36 133
87.50% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 89.26%
18 1,461 83 570 502 1,065 431 1,932 592 1,142 1,142
52.67% 92.22% 73.45% 85.81% 74.58% 86.72% 72.12% 79.14% 70.58% 20.33%
19 14 2 2 2 1 2 2 2 31 127
100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 91.37%
20 1,336 91 873 245 2,289 882 2,501 750 2,854 818 2,255
63.38% 91.92% 87.56% 94.96% 84.34% 93.14% 81.68% 88.65% 67.60% 80.27% 22.19%
21 139 19 29 42 37 26 86 57 40 136 165
65.88% 100.00% 80.56% 85.71% 77.08% 100.00% 90.53% 73.08% 80.00% 93.15% 87.30%
22 99 13 31 25 91 67 118 46 85 26 161 101
85.34% 92.86% 88.57% 96.15% 87.50% 98.53% 84.89% 95.83% 87.63% 89.66% 86.56% 49.51%
23 12 2 2 2 2 2 2 2 2 32 121
92.31% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 90.30%