The database contains 21,006,768 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.
$n \backslash r_2$ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 All
1 1 1
100.00% 0.00%
2 684,218 684,578 1,368,796
49.99% 50.01% 6.52%
3 205,352 667,539 872,891
23.53% 76.47% 4.16%
4 743,595 747,820 1,678,249 3,169,664
23.46% 23.59% 52.95% 15.09%
5 322,136 316,464 685,096 1,323,696
24.34% 23.91% 51.76% 6.30%
6 883,997 436,953 2,103,684 2,732,882 6,157,516
14.36% 7.10% 34.16% 44.38% 29.31%
7 946 23,975 436,253 702,561 1,163,735
0.08% 2.06% 37.49% 60.37% 5.54%
8 109,722 330,522 608,027 540,616 1,496,647 3,085,534
3.56% 10.71% 19.71% 17.52% 48.51% 14.69%
9 9,137 2,959 8,932 146,722 149,554 317,304
2.88% 0.93% 2.81% 46.24% 47.13% 1.51%
10 271,458 19,045 88,960 461,261 1,135,072 833,792 2,809,588
9.66% 0.68% 3.17% 16.42% 40.40% 29.68% 13.37%
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 51,939 112,208 220,852
1.62% 0.38% 2.08% 4.57% 17.02% 23.52% 50.81% 1.05%
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 39 510 1,079 2,135
12.51% 0.84% 2.48% 2.20% 5.71% 1.83% 23.89% 50.54% 0.01%
15 284 13 68 38 88 87 574 1,195 2,347
12.10% 0.55% 2.90% 1.62% 3.75% 3.71% 24.46% 50.92% 0.01%
16 5,825 115 2,685 511 12,511 1,451 12,608 2,247 35,858 73,811
7.89% 0.16% 3.64% 0.69% 16.95% 1.97% 17.08% 3.04% 48.58% 0.35%
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 4,594 90 783 588 1,449 498 80,809 754 33,246 239,106 361,917
1.27% 0.02% 0.22% 0.16% 0.40% 0.14% 22.33% 0.21% 9.19% 66.07% 1.72%
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 2,279 99 1,004 260 2,730 949 3,073 847 6,097 1,037 22,403 40,778
5.59% 0.24% 2.46% 0.64% 6.69% 2.33% 7.54% 2.08% 14.95% 2.54% 54.94% 0.19%
21 216 19 36 49 48 26 96 80 54 345 247 1,216
17.76% 1.56% 2.96% 4.03% 3.95% 2.14% 7.89% 6.58% 4.44% 28.37% 20.31% 0.01%
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 3,247,851 3,231,082 5,618,488 4,595,679 2,836,577 889,780 210,209 6,292 75,536 240,660 22,870 369

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 4,221 868,670 872,891
0.48% 99.52%
4 21,871 143,845 2,177,435 29,534 796,979 3,169,664
0.69% 4.54% 68.70% 0.93% 25.14%
5 1,004 16,194 28,830 114,826 1,162,842 1,323,696
0.08% 1.22% 2.18% 8.67% 87.85%
6 18,961 136,013 285,142 5,261 16,722 179,725 244,008 46,108 64,259 2,319 920,162 110,104 1,946,131 1,011,145 9,721 1,161,735 6,157,516
0.31% 2.21% 4.63% 0.09% 0.27% 2.92% 3.96% 0.75% 1.04% 0.04% 14.94% 1.79% 31.61% 16.42% 0.16% 18.87%
7 129 2,101 86 1,441 40,860 56,886 1,062,232 1,163,735
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
$C_n$ $=A_3$ 21,871 1,004 18,961 129 6,206 284 1,839 28 2,097 14 309 96 552 11 1,161 8 940 70 166 9
0.69% 0.08% 0.31% 0.01% 0.20% 0.09% 0.07% 1.75% 0.95% 5.09% 14.47% 4.09% 0.75% 5.05% 0.32% 4.06% 2.31% 5.76% 14.69% 4.59%
$D_n$ $=S_3$ 2,177,435 16,194 285,142 2,101 30,358 706 7,317 57 158 41 33 53 86 12 10 11 9 3 8 10
68.70% 1.22% 4.63% 0.18% 0.98% 0.22% 0.26% 3.57% 0.07% 14.91% 1.55% 2.26% 0.12% 5.50% 0.00% 5.58% 0.02% 0.25% 0.71% 5.10%
$A_n$ 4,221 29,534 114,826 9,721 56,886 125 636 1,211 79 9 5 5 5 6 3 2 1 1 2 1 2
0.48% 0.93% 8.67% 0.16% 4.89% 0.00% 0.20% 0.04% 4.95% 0.00% 1.82% 0.23% 0.21% 0.01% 1.38% 0.00% 0.51% 0.00% 0.16% 0.09% 1.02%
$S_n$ 868,670 796,979 1,162,842 1,161,735 1,062,232 4,255 3,394 59,915 1,143 162 194 160 173 171 173 165 164 165 168 168 165
99.52% 25.14% 87.85% 18.87% 91.28% 0.14% 1.07% 2.13% 71.62% 0.07% 70.55% 7.49% 7.37% 0.23% 79.36% 0.05% 83.25% 0.40% 13.82% 14.87% 84.18%

Distribution by class number

Class numbers have been computed for 99.32% 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%
$3$ 479,277 326,268 57,658 7,423 1,946 293 24 872,889
54.91% 37.38% 6.61% 0.85% 0.22% 0.03% 0.00% 100.00%
$4$ 1,045,367 896,522 445,401 427,195 271,004 81,099 2,947 90 1 3,169,626
32.98% 28.28% 14.05% 13.48% 8.55% 2.56% 0.09% 0.00% 0.00% 100.00%
$5$ 1,161,103 154,017 8,067 481 22 1 3 1 1,323,695
87.72% 11.64% 0.61% 0.04% 0.00% 0.00% 0.00% 0.00% 100.00%
$6$ 2,848,886 2,096,272 751,591 347,931 105,455 6,446 795 93 32 6 6,157,507
46.27% 34.04% 12.21% 5.65% 1.71% 0.10% 0.01% 0.00% 0.00% 0.00% 100.00%
$7$ 1,137,256 26,247 224 7 1,163,734
97.72% 2.26% 0.02% 0.00% 100.00%
$8$ 1,172,633 1,081,831 567,601 186,098 49,597 14,513 5,559 3,225 1,212 306 76 8 3,082,664
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%
$9$ 174,085 93,954 35,950 10,759 1,640 363 97 30 22 4 2 316,906
54.93% 29.65% 11.34% 3.40% 0.52% 0.11% 0.03% 0.01% 0.01% 0.00% 0.00% 99.87%
$10$ 1,802,002 839,408 124,660 38,369 1,142 608 347 121 84 14 1 1 2,806,757
64.20% 29.91% 4.44% 1.37% 0.04% 0.02% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% 99.90%
$11$ 1,482 87 25 1,594
92.97% 5.46% 1.57% 99.87%
$12$ 65,737 104,933 38,893 6,563 1,879 1,622 939 240 17 220,823
29.77% 47.52% 17.61% 2.97% 0.85% 0.73% 0.43% 0.11% 0.01% 99.99%
$13$ 264 11 275
96.00% 4.00% 100.00%
$14$ 1,035 750 138 39 44 41 37 26 8 1 2,119
48.84% 35.39% 6.51% 1.84% 2.08% 1.93% 1.75% 1.23% 0.38% 0.05% 99.25%
$15$ 1,136 453 124 19 2 1 1,735
65.48% 26.11% 7.15% 1.10% 0.12% 0.06% 73.92%
$16$ 26,356 22,463 8,506 4,891 3,889 2,677 2,085 1,382 707 307 98 24 73,418
35.90% 30.60% 11.59% 6.66% 5.30% 3.65% 2.84% 1.88% 0.96% 0.42% 0.13% 0.03% 99.47%
$17$ 199 12 6 217
91.71% 5.53% 2.76% 99.54%
$18$ 15,899 51,737 74,528 62,888 30,807 14,554 6,014 2,490 1,134 514 201 61 260,876
6.09% 19.83% 28.57% 24.11% 11.81% 5.58% 2.31% 0.95% 0.43% 0.20% 0.08% 0.02% 72.08%
$19$ 185 12 197
93.91% 6.09% 100.00%
$20$ 15,076 7,931 5,481 5,200 2,404 522 398 292 230 132 59 18 37,756
39.93% 21.01% 14.52% 13.77% 6.37% 1.38% 1.05% 0.77% 0.61% 0.35% 0.16% 0.05% 92.59%
$21$ 836 246 21 5 2 1 1,111
75.25% 22.14% 1.89% 0.45% 0.18% 0.09% 91.37%
$22$ 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 10,128,266 6,129,612 2,243,187 1,683,719 529,933 124,781 21,020 9,343 4,451 1,913 739 216 20,863,957
48.54% 29.38% 10.75% 8.07% 2.54% 0.60% 0.10% 0.04% 0.02% 0.01% 0.00% 0.00% 99.32%

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.
$n \backslash r_2$ 0 1 2 3 4 5 6 7 8 9 10 11
1 1
100.00%
2 177,159 9
25.89% 0.00%
3 156,593 322,684
76.26% 48.34%
4 460,081 372,652 212,634
61.87% 49.83% 12.67%
5 300,892 297,191 563,020
93.41% 93.91% 82.18%
6 470,296 318,016 849,230 1,211,344
53.20% 72.78% 40.37% 44.32%
7 889 23,968 425,305 687,094
93.97% 99.97% 97.49% 97.80%
8 75,380 272,805 282,737 203,800 337,911
68.70% 82.54% 46.50% 37.81% 22.60%
9 5,609 2,762 8,148 60,552 97,014
61.39% 93.34% 91.22% 41.31% 64.99%
10 208,487 15,224 50,868 302,165 727,341 497,917
76.80% 79.94% 57.18% 65.51% 64.08% 59.92%
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 13,856
35.53% 71.87% 66.72% 61.48% 50.22% 42.07% 12.35%
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 37 352 235
71.59% 100.00% 96.00% 100.00% 89.34% 94.87% 69.70% 21.88%
15 205 13 62 36 76 75 278 391
74.55% 100.00% 91.18% 94.74% 86.36% 88.24% 69.67% 50.85%
16 3,145 109 2,071 461 7,174 988 5,640 1,022 5,746
54.17% 94.78% 77.19% 90.22% 57.57% 68.14% 44.77% 45.50% 16.16%
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 1,649 83 570 502 1,065 431 5,031 592 2,152 3,824
41.06% 92.22% 73.45% 85.81% 74.58% 86.72% 7.96% 79.14% 7.43% 2.38%
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 1,340 91 873 245 2,289 882 2,501 750 2,936 818 2,351
62.56% 91.92% 87.56% 94.96% 84.34% 93.14% 81.68% 88.65% 57.31% 80.04% 11.44%
21 139 19 29 42 37 26 86 57 40 192 169
65.88% 100.00% 80.56% 85.71% 77.08% 100.00% 90.53% 73.08% 80.00% 63.58% 85.79%
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%