The database contains 19,799,405 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 All
1 1 1
100.00% 0.00%
2 684,218 684,578 1,368,796
49.99% 50.01% 6.91%
3 205,160 663,362 868,522
23.62% 76.38% 4.39%
4 703,725 674,910 1,578,112 2,956,747
23.80% 22.83% 53.37% 14.93%
5 321,560 312,076 675,305 1,308,941
24.57% 23.84% 51.59% 6.61%
6 852,347 436,953 2,086,940 2,372,103 5,748,343
14.83% 7.60% 36.31% 41.27% 29.03%
7 935 23,975 436,253 702,543 1,163,706
0.08% 2.06% 37.49% 60.37% 5.88%
8 108,774 330,522 608,027 540,615 1,493,182 3,081,120
3.53% 10.73% 19.73% 17.55% 48.46% 15.56%
9 8,419 2,959 8,932 92,923 121,792 235,025
3.58% 1.26% 3.80% 39.54% 51.82% 1.19%
10 261,976 9,083 79,530 451,556 1,124,403 820,671 2,747,219
9.54% 0.33% 2.89% 16.44% 40.93% 29.87% 13.88%
11 44 6 8 19 636 873 1,586
2.77% 0.38% 0.50% 1.20% 40.10% 55.04% 0.01%
12 1,564 846 4,604 10,091 37,595 51,939 105,873 212,512
0.74% 0.40% 2.17% 4.75% 17.69% 24.44% 49.82% 1.07%
13 19 2 2 2 10 51 180 266
7.14% 0.75% 0.75% 0.75% 3.76% 19.17% 67.67% 0.00%
14 144 18 53 47 122 39 259 610 1,292
11.15% 1.39% 4.10% 3.64% 9.44% 3.02% 20.05% 47.21% 0.01%
15 206 13 68 38 88 82 221 247 963
21.39% 1.35% 7.06% 3.95% 9.14% 8.52% 22.95% 25.65% 0.00%
16 3,991 115 2,685 511 12,511 1,451 12,608 2,247 25,689 61,808
6.46% 0.19% 4.34% 0.83% 20.24% 2.35% 20.40% 3.64% 41.56% 0.31%
17 10 2 2 1 2 2 7 36 149 211
4.74% 0.95% 0.95% 0.47% 0.95% 0.95% 3.32% 17.06% 70.62% 0.00%
18 1,988 90 783 588 1,449 498 2,720 754 1,658 4,732 15,260
13.03% 0.59% 5.13% 3.85% 9.50% 3.26% 17.82% 4.94% 10.87% 31.01% 0.08%
19 9 2 2 2 1 2 2 2 31 139 192
4.69% 1.04% 1.04% 1.04% 0.52% 1.04% 1.04% 1.04% 16.15% 72.40% 0.00%
20 1,548 99 1,004 260 2,730 949 3,073 847 4,248 1,023 9,018 24,799
6.24% 0.40% 4.05% 1.05% 11.01% 3.83% 12.39% 3.42% 17.13% 4.13% 36.36% 0.13%
21 152 19 36 49 48 26 96 80 54 149 191 900
16.89% 2.11% 4.00% 5.44% 5.33% 2.89% 10.67% 8.89% 6.00% 16.56% 21.22% 0.00%
22 71 15 38 27 110 69 140 50 99 31 188 169 1,007
7.05% 1.49% 3.77% 2.68% 10.92% 6.85% 13.90% 4.97% 9.83% 3.08% 18.67% 16.78% 0.01%
23 7 0 2 2 2 2 2 2 2 2 32 134 189
3.70% 0.00% 1.06% 1.06% 1.06% 1.06% 1.06% 1.06% 1.06% 1.06% 16.93% 70.90% 0.00%
Total 3,156,868 3,139,645 5,482,386 4,171,377 2,794,681 876,654 125,181 4,875 31,930 6,076 9,429 303

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 21,596 117,034 1,991,604 29,534 796,979 2,956,747
0.73% 3.96% 67.36% 1.00% 26.95%
5 983 15,955 26,986 113,768 1,151,249 1,308,941
0.08% 1.22% 2.06% 8.69% 87.95%
6 15,543 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 5,748,343
0.27% 0.43% 4.13% 0.09% 0.25% 3.11% 4.24% 0.80% 1.12% 0.04% 11.78% 1.92% 33.86% 17.59% 0.17% 20.21%
7 118 2,101 86 1,423 40,860 56,886 1,062,232 1,163,706
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
$C_n$ $=A_3$ 21,596 983 15,543 118 5,817 218 1,464 18 1,059 5 38 20 117 4 143 3 69 6 43 2
0.73% 0.08% 0.27% 0.01% 0.19% 0.09% 0.05% 1.13% 0.50% 1.88% 2.94% 2.08% 0.19% 1.90% 0.94% 1.56% 0.28% 0.67% 4.27% 1.06%
$D_n$ $=S_3$ 1,991,604 15,955 237,228 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% 0.99% 0.30% 0.27% 3.59% 0.07% 15.41% 2.55% 5.50% 0.14% 5.69% 0.07% 5.73% 0.04% 0.33% 0.79% 5.29%
$A_n$ 4,029 29,534 113,768 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% 8.69% 0.17% 4.89% 0.00% 0.27% 0.04% 4.98% 0.00% 1.88% 0.39% 0.52% 0.01% 1.42% 0.01% 0.52% 0.00% 0.22% 0.10% 1.06%
$S_n$ 864,493 796,979 1,151,249 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% 87.95% 20.21% 91.28% 0.14% 1.44% 0.19% 72.07% 0.08% 72.93% 12.38% 17.96% 0.28% 81.99% 1.08% 85.42% 0.67% 18.67% 16.68% 87.30%

Distribution by class number

Class numbers have been computed for 99.98% 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%
$4$ 1,028,892 812,963 375,912 389,483 265,607 80,826 2,935 90 1 2,956,709
34.80% 27.50% 12.71% 13.17% 8.98% 2.73% 0.10% 0.00% 0.00% 100.00%
$5$ 1,151,513 149,607 7,371 430 19 1 1,308,941
87.97% 11.43% 0.56% 0.03% 0.00% 0.00% 100.00%
$6$ 2,837,357 2,052,716 643,803 160,964 51,570 1,919 8 5,748,337
49.36% 35.71% 11.20% 2.80% 0.90% 0.03% 0.00% 100.00%
$7$ 1,137,252 26,231 218 5 1,163,706
97.73% 2.25% 0.02% 0.00% 100.00%
$8$ 1,172,612 1,081,304 567,130 185,078 47,897 13,892 5,505 3,225 1,212 306 76 8 3,078,250
38.09% 35.13% 18.42% 6.01% 1.56% 0.45% 0.18% 0.10% 0.04% 0.01% 0.00% 0.00% 99.91%
$9$ 168,996 61,456 4,445 128 235,025
71.91% 26.15% 1.89% 0.05% 100.00%
$10$ 1,745,184 838,193 123,929 38,124 1,039 495 246 2,747,210
63.53% 30.51% 4.51% 1.39% 0.04% 0.02% 0.01% 100.00%
$11$ 1,482 87 15 1,584
93.56% 5.49% 0.95% 99.87%
$12$ 65,437 103,430 37,863 5,222 266 176 88 2 212,484
30.80% 48.68% 17.82% 2.46% 0.13% 0.08% 0.04% 0.00% 99.99%
$13$ 255 11 266
95.86% 4.14% 100.00%
$14$ 770 436 52 18 10 1,286
59.88% 33.90% 4.04% 1.40% 0.78% 99.54%
$15$ 836 109 7 2 954
87.63% 11.43% 0.73% 0.21% 99.07%
$16$ 26,093 21,092 7,130 3,409 2,033 920 463 249 152 76 43 21 61,713
42.28% 34.18% 11.55% 5.52% 3.29% 1.49% 0.75% 0.40% 0.25% 0.12% 0.07% 0.03% 99.85%
$17$ 192 12 6 210
91.43% 5.71% 2.86% 99.53%
$18$ 8,825 3,966 1,058 810 277 65 32 1 1 15,036
58.69% 26.38% 7.04% 5.39% 1.84% 0.43% 0.21% 0.01% 0.01% 98.53%
$19$ 180 12 192
93.75% 6.25% 100.00%
$20$ 14,796 6,299 2,080 659 479 202 104 32 22 9 2 2 24,691
59.92% 25.51% 8.42% 2.67% 1.94% 0.82% 0.42% 0.13% 0.09% 0.04% 0.01% 0.01% 99.56%
$21$ 750 135 1 886
84.65% 15.24% 0.11% 98.44%
$22$ 827 105 16 4 4 7 12 4 979
84.47% 10.73% 1.63% 0.41% 0.41% 0.72% 1.23% 0.41% 97.22%
$23$ 174 13 1 188
92.55% 6.91% 0.53% 99.47%
Total 10,018,535 5,907,227 1,950,691 1,376,178 429,556 98,782 9,418 3,603 1,387 392 121 31 19,795,964
50.61% 29.84% 9.85% 6.95% 2.17% 0.50% 0.05% 0.02% 0.01% 0.00% 0.00% 0.00% 99.98%

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%
3 156,575 322,368
76.32% 48.60%
4 452,363 364,005 212,524
64.28% 53.93% 13.47%
5 300,434 293,565 557,514
93.43% 94.07% 82.56%
6 464,517 318,016 847,759 1,207,065
54.50% 72.78% 40.62% 50.89%
7 889 23,968 425,305 687,090
95.08% 99.97% 97.49% 97.80%
8 75,360 272,805 282,737 203,799 337,911
69.28% 82.54% 46.50% 37.81% 22.65%
9 5,480 2,762 8,148 57,148 95,458
65.09% 93.34% 91.22% 61.50% 78.38%
10 199,179 5,262 41,438 292,460 717,497 489,348
76.03% 57.93% 52.10% 64.77% 63.81% 59.63%
11 42 6 8 19 609 798
100.00% 100.00% 100.00% 100.00% 95.75% 91.41%
12 998 608 3,072 6,204 18,880 21,849 13,826
63.81% 71.87% 66.72% 61.48% 50.22% 42.07% 13.06%
13 19 2 2 2 10 50 170
100.00% 100.00% 100.00% 100.00% 100.00% 98.04% 94.44%
14 125 18 48 47 109 37 225 161
88.65% 100.00% 96.00% 100.00% 89.34% 94.87% 86.87% 26.39%
15 168 13 62 36 76 74 202 205
85.28% 100.00% 91.18% 94.74% 86.36% 90.24% 91.40% 83.00%
16 2,913 109 2,071 461 7,174 988 5,640 1,022 5,715
73.10% 94.78% 77.19% 90.22% 57.57% 68.14% 44.77% 45.50% 22.27%
17 7 2 2 1 2 2 7 36 133
77.78% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 89.26%
18 1,366 83 570 502 1,065 431 1,932 592 1,142 1,142
70.56% 92.22% 73.45% 85.81% 74.58% 86.72% 72.12% 79.14% 70.58% 24.41%
19 9 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 1,238 91 873 245 2,289 882 2,501 750 2,854 818 2,255
80.86% 91.92% 87.56% 94.96% 84.34% 93.14% 81.68% 88.65% 67.60% 80.27% 25.07%
21 113 19 29 42 37 26 86 57 40 136 165
75.33% 100.00% 80.56% 85.71% 77.08% 100.00% 90.53% 73.08% 80.00% 93.15% 87.30%
22 63 13 31 25 91 67 118 46 85 26 161 101
94.03% 92.86% 88.57% 96.15% 87.50% 98.53% 84.89% 95.83% 87.63% 89.66% 86.56% 60.84%
23 5 2 2 2 2 2 2 2 2 32 121
83.33% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 90.30%