The database contains 18,683,677 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%
2 684,216 684,128 1,368,344
50.00% 50.00% 7.32%
3 201,440 657,369 858,809
23.46% 76.54% 4.60%
4 672,184 595,920 1,070,120 2,338,224
28.75% 25.49% 45.77% 12.51%
5 274,956 312,070 675,285 1,262,311
21.78% 24.72% 53.50% 6.76%
6 828,871 436,909 2,086,754 2,170,066 5,522,600
15.01% 7.91% 37.79% 39.29% 29.56%
7 932 18,303 369,332 574,036 962,603
0.10% 1.90% 38.37% 59.63% 5.15%
8 107,694 330,500 607,204 540,440 1,492,206 3,078,044
3.50% 10.74% 19.73% 17.56% 48.48% 16.47%
9 8,378 2,945 8,918 92,873 121,668 234,782
3.57% 1.25% 3.80% 39.56% 51.82% 1.26%
10 261,472 8,909 79,025 451,369 1,120,951 817,265 2,738,991
9.55% 0.33% 2.89% 16.48% 40.93% 29.84% 14.66%
11 35 2 3 14 599 748 1,401
2.50% 0.14% 0.21% 1.00% 42.76% 53.39% 0.01%
12 1,564 846 4,604 10,091 37,595 51,836 105,826 212,362
0.74% 0.40% 2.17% 4.75% 17.70% 24.41% 49.83% 1.14%
13 19 2 2 2 10 23 58 116
16.38% 1.72% 1.72% 1.72% 8.62% 19.83% 50.00% 0.00%
14 144 18 53 47 122 39 153 560 1,136
12.68% 1.58% 4.67% 4.14% 10.74% 3.43% 13.47% 49.30% 0.01%
15 206 13 68 38 88 82 193 125 813
25.34% 1.60% 8.36% 4.67% 10.82% 10.09% 23.74% 15.38% 0.00%
16 3,991 115 2,685 511 12,511 1,451 12,608 2,141 25,639 61,652
6.47% 0.19% 4.36% 0.83% 20.29% 2.35% 20.45% 3.47% 41.59% 0.33%
17 10 2 2 1 2 2 7 8 28 62
16.13% 3.23% 3.23% 1.61% 3.23% 3.23% 11.29% 12.90% 45.16% 0.00%
18 1,988 90 783 588 1,449 498 2,720 754 1,552 4,680 15,102
13.16% 0.60% 5.18% 3.89% 9.59% 3.30% 18.01% 4.99% 10.28% 30.99% 0.08%
19 9 2 2 2 1 2 2 2 3 17 42
21.43% 4.76% 4.76% 4.76% 2.38% 4.76% 4.76% 4.76% 7.14% 40.48% 0.00%
20 1,548 99 1,004 260 2,730 949 3,073 847 4,248 917 8,968 24,643
6.28% 0.40% 4.07% 1.06% 11.08% 3.85% 12.47% 3.44% 17.24% 3.72% 36.39% 0.13%
21 152 19 36 49 48 26 96 80 54 121 68 749
20.29% 2.54% 4.81% 6.54% 6.41% 3.47% 12.82% 10.68% 7.21% 16.15% 9.08% 0.00%
22 71 15 38 27 110 69 140 50 99 31 82 118 850
8.35% 1.76% 4.47% 3.18% 12.94% 8.12% 16.47% 5.88% 11.65% 3.65% 9.65% 13.88% 0.00%
23 7 0 2 2 2 2 2 2 2 2 4 13 40
17.50% 0.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 10.00% 32.50% 0.00%
Total 3,049,888 3,048,276 4,905,920 3,840,416 2,790,092 872,992 124,878 4,569 31,625 5,768 9,122 131

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 1,822 856,987 858,809
0.21% 99.79%
4 11,945 99,475 1,477,849 28,787 720,168 2,338,224
0.51% 4.25% 63.20% 1.23% 30.80%
5 983 15,955 26,984 113,768 1,104,621 1,262,311
0.08% 1.26% 2.14% 9.01% 87.51%
6 15,542 24,493 236,290 5,261 14,611 177,913 244,006 46,108 64,234 2,312 476,289 110,104 1,946,055 1,011,145 9,721 1,138,516 5,522,600
0.28% 0.44% 4.28% 0.10% 0.26% 3.22% 4.42% 0.83% 1.16% 0.04% 8.62% 1.99% 35.24% 18.31% 0.18% 20.62%
7 118 2,101 86 1,423 40,860 56,883 861,132 962,603
0.01% 0.22% 0.01% 0.15% 4.24% 5.91% 89.46%

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
$C_n$ $=A_3$ 11,945 983 15,542 118 5,817 218 1,461 18 1,059 5 38 20 117 4 143 3 69 6 43 2
0.51% 0.08% 0.28% 0.01% 0.19% 0.09% 0.05% 1.28% 0.50% 4.31% 3.35% 2.46% 0.19% 6.45% 0.95% 7.14% 0.28% 0.80% 5.06% 5.00%
$D_n$ $=S_3$ 1,477,849 15,955 236,290 2,101 30,358 706 885 56 158 41 33 53 86 12 10 11 9 3 8 10
63.20% 1.26% 4.28% 0.22% 0.99% 0.30% 0.03% 4.00% 0.07% 35.34% 2.90% 6.52% 0.14% 19.35% 0.07% 26.19% 0.04% 0.40% 0.94% 25.00%
$A_n$ 1,822 28,787 113,768 9,721 56,883 122 633 1,209 76 9 5 5 5 6 3 2 1 1 2 1 2
0.21% 1.23% 9.01% 0.18% 5.91% 0.00% 0.27% 0.04% 5.42% 0.00% 4.31% 0.44% 0.62% 0.01% 4.84% 0.01% 2.38% 0.00% 0.27% 0.12% 5.00%
$S_n$ 856,987 720,168 1,104,621 1,138,516 861,132 4,101 3,230 4,977 968 16 48 16 28 25 28 19 18 20 22 22 20
99.79% 30.80% 87.51% 20.62% 89.46% 0.13% 1.38% 0.18% 69.09% 0.01% 41.38% 1.41% 3.44% 0.04% 45.16% 0.13% 42.86% 0.08% 2.94% 2.59% 50.00%

Distribution by class number

Class numbers have been computed for 99.94% 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
$2$ 177,168 424,902 123,439 584,569 58,266 1,368,344
12.95% 31.05% 9.02% 42.72% 4.26% 100.00%
$3$ 477,929 317,813 54,072 6,810 1,881 280 24 858,809
55.65% 37.01% 6.30% 0.79% 0.22% 0.03% 0.00% 100.00%
$4$ 1,005,496 732,254 309,624 232,952 46,547 9,741 1,520 88 1 2,338,223
43.00% 31.32% 13.24% 9.96% 1.99% 0.42% 0.07% 0.00% 0.00% 100.00%
$5$ 1,106,122 148,368 7,371 430 19 1 1,262,311
87.63% 11.75% 0.58% 0.03% 0.00% 0.00% 100.00%
$6$ 2,814,061 2,048,077 594,480 57,341 7,293 1,342 6 5,522,600
50.96% 37.09% 10.76% 1.04% 0.13% 0.02% 0.00% 100.00%
$7$ 939,652 22,728 218 5 962,603
97.62% 2.36% 0.02% 0.00% 100.00%
$8$ 1,171,123 1,077,994 565,180 183,829 47,254 13,695 5,381 3,107 1,177 300 76 8 3,069,129
38.16% 35.12% 18.41% 5.99% 1.54% 0.45% 0.18% 0.10% 0.04% 0.01% 0.00% 0.00% 99.71%
$9$ 168,777 61,432 4,445 128 234,782
71.89% 26.17% 1.89% 0.05% 100.00%
$10$ 1,740,922 834,230 122,689 37,962 1,018 492 244 2,737,557
63.59% 30.47% 4.48% 1.39% 0.04% 0.02% 0.01% 99.95%
$11$ 1,311 73 15 1,399
93.71% 5.22% 1.07% 99.86%
$12$ 65,316 103,402 37,862 5,222 266 176 88 2 212,334
30.76% 48.70% 17.83% 2.46% 0.13% 0.08% 0.04% 0.00% 99.99%
$13$ 116 116
100.00% 100.00%
$14$ 642 408 52 18 10 1,130
56.81% 36.11% 4.60% 1.59% 0.88% 99.47%
$15$ 702 93 7 2 804
87.31% 11.57% 0.87% 0.25% 98.89%
$16$ 25,964 21,065 7,130 3,409 2,033 920 463 249 152 76 43 21 61,557
42.18% 34.22% 11.58% 5.54% 3.30% 1.49% 0.75% 0.40% 0.25% 0.12% 0.07% 0.03% 99.85%
$17$ 55 6 61
90.16% 9.84% 98.39%
$18$ 8,688 3,939 1,058 810 277 65 32 1 1 14,872
58.42% 26.49% 7.11% 5.45% 1.86% 0.44% 0.22% 0.01% 0.01% 98.48%
$19$ 42 42
100.00% 100.00%
$20$ 14,667 6,272 2,080 659 479 202 104 32 22 9 2 2 24,535
59.78% 25.56% 8.48% 2.69% 1.95% 0.82% 0.42% 0.13% 0.09% 0.04% 0.01% 0.01% 99.56%
$21$ 611 123 1 735
83.13% 16.73% 0.14% 98.13%
$22$ 691 84 16 4 4 7 12 4 822
84.06% 10.22% 1.95% 0.49% 0.49% 0.85% 1.46% 0.49% 96.71%
$23$ 38 1 39
97.44% 2.56% 97.50%
Total 9,720,094 5,803,257 1,829,746 1,114,150 165,347 26,920 7,875 3,483 1,352 386 121 31 18,672,805
52.05% 31.08% 9.80% 5.97% 0.89% 0.14% 0.04% 0.02% 0.01% 0.00% 0.00% 0.00% 99.94%

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%
3 155,974 321,955
77.43% 48.98%
4 441,080 352,625 211,791
65.62% 59.17% 19.79%
5 255,061 293,560 557,501
92.76% 94.07% 82.56%
6 441,394 317,987 847,643 1,207,037
53.25% 72.78% 40.62% 55.62%
7 886 18,296 358,558 561,912
95.06% 99.96% 97.08% 97.89%
8 74,764 272,695 282,313 203,536 337,815
69.56% 82.62% 46.55% 37.92% 22.70%
9 5,445 2,749 8,134 57,102 95,347
64.99% 93.34% 91.21% 61.48% 78.37%
10 198,292 5,109 41,039 292,308 715,689 488,485
76.19% 57.35% 51.93% 64.76% 63.85% 59.78%
11 33 2 3 14 572 687
100.00% 100.00% 100.00% 100.00% 95.49% 91.84%
12 998 608 3,072 6,204 18,880 21,757 13,797
63.81% 71.87% 66.72% 61.48% 50.22% 41.97% 13.04%
13 19 2 2 2 10 23 58
100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00%
14 125 18 48 47 109 37 129 129
88.65% 100.00% 96.00% 100.00% 89.34% 94.87% 84.31% 23.04%
15 168 13 62 36 76 74 174 99
85.28% 100.00% 91.18% 94.74% 86.36% 90.24% 90.16% 79.20%
16 2,913 109 2,071 461 7,174 988 5,640 925 5,683
73.10% 94.78% 77.19% 90.22% 57.57% 68.14% 44.77% 43.22% 22.19%
17 7 2 2 1 2 2 7 8 24
77.78% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 85.71%
18 1,365 83 570 502 1,063 431 1,932 592 1,045 1,105
70.54% 92.22% 73.55% 85.81% 74.60% 86.72% 72.12% 79.14% 69.11% 23.89%
19 9 2 2 2 1 2 2 2 3 17
100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00%
20 1,238 91 873 245 2,289 882 2,501 750 2,854 722 2,222
80.86% 91.92% 87.56% 94.96% 84.34% 93.14% 81.68% 88.65% 67.60% 79.08% 24.84%
21 113 19 29 42 37 26 86 57 40 108 54
75.33% 100.00% 80.56% 85.71% 77.08% 100.00% 90.53% 73.08% 80.00% 91.53% 81.82%
22 63 13 31 25 91 67 118 46 85 26 62 64
94.03% 92.86% 88.57% 96.15% 87.50% 98.53% 84.89% 95.83% 87.63% 89.66% 77.50% 55.65%
23 5 2 2 2 2 2 2 2 2 4 13
83.33% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00%