The database contains 18,686,376 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%
5 274,956 312,074 675,304 1,262,334
21.78% 24.72% 53.50% 6.76%
6 828,871 436,909 2,086,823 2,170,098 5,522,701
15.01% 7.91% 37.79% 39.29% 29.55%
7 932 18,303 369,357 574,135 962,727
0.10% 1.90% 38.37% 59.64% 5.15%
8 107,694 330,500 607,204 540,563 1,492,262 3,078,223
3.50% 10.74% 19.73% 17.56% 48.48% 16.47%
9 8,378 2,945 8,918 92,901 121,785 234,927
3.57% 1.25% 3.80% 39.54% 51.84% 1.26%
10 261,472 8,909 79,025 451,369 1,121,052 817,310 2,739,137
9.55% 0.33% 2.89% 16.48% 40.93% 29.84% 14.66%
11 35 2 3 14 627 869 1,550
2.26% 0.13% 0.19% 0.90% 40.45% 56.06% 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% 1.14%
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% 0.01%
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% 0.33%
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 3,049,888 3,048,280 4,906,033 3,840,698 2,790,394 873,289 125,181 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 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 1,104,644 1,262,334
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 1,138,617 5,522,701
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 56,883 861,256 962,727
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% 1.16% 0.50% 1.88% 2.94% 2.08% 0.19% 1.90% 0.94% 1.56% 0.28% 0.67% 4.27% 1.06%
$D_n$ $=S_3$ 1,477,849 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% 3.61% 0.07% 15.41% 2.55% 5.50% 0.14% 5.69% 0.07% 5.73% 0.04% 0.33% 0.79% 5.29%
$A_n$ 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% 4.90% 0.00% 1.88% 0.39% 0.52% 0.01% 1.42% 0.01% 0.52% 0.00% 0.22% 0.10% 1.06%
$S_n$ 856,987 720,168 1,104,644 1,138,617 861,256 4,242 3,375 5,123 1,113 162 194 160 173 171 173 165 164 165 168 168 165
99.79% 30.80% 87.51% 20.62% 89.46% 0.14% 1.44% 0.19% 71.81% 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
$2$ 177,168 424,902 123,439 584,569 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%
$4$ 1,005,496 732,254 309,624 232,953 46,547 9,741 1,520 88 1 2,338,224
43.00% 31.32% 13.24% 9.96% 1.99% 0.42% 0.07% 0.00% 0.00% 100.00%
$5$ 1,106,137 148,376 7,371 430 19 1 1,262,334
87.63% 11.75% 0.58% 0.03% 0.00% 0.00% 100.00%
$6$ 2,814,135 2,048,104 594,480 57,341 7,293 1,342 6 5,522,701
50.96% 37.09% 10.76% 1.04% 0.13% 0.02% 0.00% 100.00%
$7$ 939,760 22,744 218 5 962,727
97.61% 2.36% 0.02% 0.00% 100.00%
$8$ 1,171,695 1,080,016 566,742 184,871 47,841 13,871 5,497 3,221 1,212 306 76 8 3,075,361
38.10% 35.12% 18.43% 6.01% 1.56% 0.45% 0.18% 0.10% 0.04% 0.01% 0.00% 0.00% 99.91%
$9$ 168,909 61,445 4,445 128 234,927
71.90% 26.15% 1.89% 0.05% 100.00%
$10$ 1,741,618 835,070 122,724 37,965 1,018 495 246 2,739,136
63.58% 30.49% 4.48% 1.39% 0.04% 0.02% 0.01% 100.00%
$11$ 1,446 87 15 1,548
93.41% 5.62% 0.97% 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 9,723,429 5,806,431 1,831,344 1,115,196 165,934 27,099 7,993 3,597 1,387 392 121 31 18,682,997
52.04% 31.08% 9.80% 5.97% 0.89% 0.15% 0.04% 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%
3 155,974 321,955
77.43% 48.98%
4 441,080 352,625 211,791
65.62% 59.17% 19.79%
5 255,061 293,563 557,513
92.76% 94.07% 82.56%
6 441,394 317,987 847,703 1,207,051
53.25% 72.78% 40.62% 55.62%
7 886 18,296 358,580 561,998
95.06% 99.96% 97.08% 97.89%
8 74,830 272,792 282,404 203,784 337,885
69.48% 82.54% 46.51% 37.81% 22.66%
9 5,445 2,749 8,134 57,130 95,451
64.99% 93.34% 91.21% 61.50% 78.38%
10 198,791 5,109 41,040 292,312 715,830 488,536
76.03% 57.35% 51.93% 64.76% 63.85% 59.77%
11 33 2 3 14 600 794
100.00% 100.00% 100.00% 100.00% 95.69% 91.37%
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%