The database contains 21,461,062 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%
2 685,158 685,501 1,370,659
49.99% 50.01% 6.39%
3 266,649 913,287 1,179,936
22.60% 77.40% 5.50%
4 743,786 748,619 1,678,344 3,170,749
23.46% 23.61% 52.93% 14.77%
5 322,140 317,423 686,902 1,326,465
24.29% 23.93% 51.78% 6.18%
6 886,745 438,679 2,119,885 2,766,490 6,211,799
14.28% 7.06% 34.13% 44.54% 28.94%
7 946 23,975 436,253 702,701 1,163,875
0.08% 2.06% 37.48% 60.38% 5.42%
8 112,998 330,522 608,035 545,629 1,500,997 3,098,181
3.65% 10.67% 19.63% 17.61% 48.45% 14.44%
9 9,522 2,959 8,932 170,954 160,576 352,943
2.70% 0.84% 2.53% 48.44% 45.50% 1.64%
10 280,367 19,054 89,046 461,424 1,136,819 841,803 2,828,513
9.91% 0.67% 3.15% 16.31% 40.19% 29.76% 13.18%
11 54 6 8 19 636 873 1,596
3.38% 0.38% 0.50% 1.19% 39.85% 54.70% 0.01%
12 4,252 846 4,609 10,091 37,754 52,025 114,450 224,027
1.90% 0.38% 2.06% 4.50% 16.85% 23.22% 51.09% 1.04%
13 29 2 2 2 10 51 216 312
9.29% 0.64% 0.64% 0.64% 3.21% 16.35% 69.23% 0.00%
14 269 18 53 47 122 39 709 1,322 2,579
10.43% 0.70% 2.06% 1.82% 4.73% 1.51% 27.49% 51.26% 0.01%
15 284 13 68 38 88 96 575 1,303 2,465
11.52% 0.53% 2.76% 1.54% 3.57% 3.89% 23.33% 52.86% 0.01%
16 8,832 115 2,685 511 12,511 1,451 12,608 2,247 46,427 87,387
10.11% 0.13% 3.07% 0.58% 14.32% 1.66% 14.43% 2.57% 53.13% 0.41%
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 5,102 90 783 588 1,449 498 80,903 754 33,246 239,893 363,306
1.40% 0.02% 0.22% 0.16% 0.40% 0.14% 22.27% 0.21% 9.15% 66.03% 1.69%
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 1,004 268 2,730 953 3,079 861 6,100 1,039 22,406 40,818
5.58% 0.24% 2.46% 0.66% 6.69% 2.33% 7.54% 2.11% 14.94% 2.55% 54.89% 0.19%
21 216 19 36 49 48 26 96 81 54 345 247 1,217
17.75% 1.56% 2.96% 4.03% 3.94% 2.14% 7.89% 6.66% 4.44% 28.35% 20.30% 0.01%
22 128 15 38 27 110 69 140 50 100 31 188 236 1,132
11.31% 1.33% 3.36% 2.39% 9.72% 6.10% 12.37% 4.42% 8.83% 2.74% 16.61% 20.85% 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 3,329,802 3,481,246 5,636,689 4,658,843 2,853,855 897,890 212,787 6,658 86,109 241,449 22,873 370

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 4,678 1,175,258 1,179,936
0.40% 99.60%
4 21,871 143,933 2,177,637 29,534 797,774 3,170,749
0.69% 4.54% 68.68% 0.93% 25.16%
5 1,008 16,306 29,433 114,905 1,164,813 1,326,465
0.08% 1.23% 2.22% 8.66% 87.81%
6 21,145 136,022 320,818 5,261 22,119 179,812 244,008 46,108 65,311 2,319 923,486 110,157 1,946,133 1,011,145 9,721 1,168,234 6,211,799
0.34% 2.19% 5.16% 0.08% 0.36% 2.89% 3.93% 0.74% 1.05% 0.04% 14.87% 1.77% 31.33% 16.28% 0.16% 18.81%
7 129 2,135 86 1,547 40,860 56,886 1,062,232 1,163,875
0.01% 0.18% 0.01% 0.13% 3.51% 4.89% 91.27%

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 1,008 21,145 129 6,206 284 1,854 28 2,098 14 325 96 552 11 1,161 8 940 70 166 9
0.69% 0.08% 0.34% 0.01% 0.20% 0.08% 0.07% 1.75% 0.94% 4.49% 12.60% 3.89% 0.63% 5.05% 0.32% 4.06% 2.30% 5.75% 14.66% 4.59%
$D_n$ $=S_3$ 2,177,637 16,306 320,818 2,135 30,398 737 7,457 57 302 78 426 161 86 12 10 11 9 3 8 10
68.68% 1.23% 5.16% 0.18% 0.98% 0.21% 0.26% 3.57% 0.13% 25.00% 16.52% 6.53% 0.10% 5.50% 0.00% 5.58% 0.02% 0.25% 0.71% 5.10%
$A_n$ 4,678 29,534 114,905 9,721 56,886 125 636 1,211 79 9 5 5 5 6 3 2 1 1 2 1 2
0.40% 0.93% 8.66% 0.16% 4.89% 0.00% 0.18% 0.04% 4.95% 0.00% 1.60% 0.19% 0.20% 0.01% 1.38% 0.00% 0.51% 0.00% 0.16% 0.09% 1.02%
$S_n$ 1,175,258 797,774 1,164,813 1,168,234 1,062,232 4,255 3,394 59,915 1,143 162 194 160 173 171 173 165 164 165 168 168 165
99.60% 25.16% 87.81% 18.81% 91.27% 0.14% 0.96% 2.12% 71.62% 0.07% 62.18% 6.20% 7.02% 0.20% 79.36% 0.05% 83.25% 0.40% 13.80% 14.84% 84.18%

Distribution by class number

Class numbers have been computed for 99.01% 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
$2$ 177,168 424,929 124,112 585,283 59,025 136 6 1,370,659
12.93% 31.00% 9.05% 42.70% 4.31% 0.01% 0.00% 100.00%
$3$ 484,192 389,451 146,404 61,199 21,954 5,992 1,160 154 23 2 1,110,531
43.60% 35.07% 13.18% 5.51% 1.98% 0.54% 0.10% 0.01% 0.00% 0.00% 94.12%
$4$ 1,045,998 896,782 445,526 427,257 271,007 81,101 2,949 90 1 3,170,711
32.99% 28.28% 14.05% 13.48% 8.55% 2.56% 0.09% 0.00% 0.00% 100.00%
$5$ 1,162,917 154,722 8,287 508 24 1 3 2 1,326,464
87.67% 11.66% 0.62% 0.04% 0.00% 0.00% 0.00% 0.00% 100.00%
$6$ 2,855,371 2,102,553 759,790 361,015 117,337 12,319 2,432 643 271 48 6 1 6,211,786
45.97% 33.85% 12.23% 5.81% 1.89% 0.20% 0.04% 0.01% 0.00% 0.00% 0.00% 0.00% 100.00%
$7$ 1,137,311 26,310 245 7 1 1,163,874
97.72% 2.26% 0.02% 0.00% 0.00% 100.00%
$8$ 1,177,772 1,085,410 569,475 187,734 49,990 14,538 5,560 3,225 1,212 306 76 8 3,095,311
38.05% 35.07% 18.40% 6.07% 1.62% 0.47% 0.18% 0.10% 0.04% 0.01% 0.00% 0.00% 99.91%
$9$ 175,144 103,509 50,462 18,790 3,463 761 217 75 48 19 15 10 352,514
49.68% 29.36% 14.31% 5.33% 0.98% 0.22% 0.06% 0.02% 0.01% 0.01% 0.00% 0.00% 99.88%
$10$ 1,811,249 841,718 129,866 40,412 1,213 623 352 124 84 14 1 1 2,825,657
64.10% 29.79% 4.60% 1.43% 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%
$12$ 66,347 105,727 39,708 7,190 2,139 1,685 945 240 17 223,998
29.62% 47.20% 17.73% 3.21% 0.95% 0.75% 0.42% 0.11% 0.01% 99.99%
$13$ 268 11 33 312
85.90% 3.53% 10.58% 100.00%
$14$ 1,276 876 183 53 46 46 46 27 9 1 2,563
49.79% 34.18% 7.14% 2.07% 1.79% 1.79% 1.79% 1.05% 0.35% 0.04% 99.38%
$15$ 1,146 487 189 28 2 1 1,853
61.85% 26.28% 10.20% 1.51% 0.11% 0.05% 75.17%
$16$ 28,965 23,490 10,781 8,831 6,720 3,501 2,151 1,382 707 307 98 24 86,990
33.30% 27.00% 12.39% 10.15% 7.73% 4.02% 2.47% 1.59% 0.81% 0.35% 0.11% 0.03% 99.55%
$17$ 199 12 6 217
91.71% 5.53% 2.76% 99.54%
$18$ 16,489 51,835 74,637 63,129 31,081 14,629 6,015 2,490 1,134 514 201 61 262,264
6.29% 19.76% 28.46% 24.07% 11.85% 5.58% 2.29% 0.95% 0.43% 0.20% 0.08% 0.02% 72.19%
$19$ 185 12 197
93.91% 6.09% 100.00%
$20$ 15,089 7,942 5,481 5,200 2,404 522 398 292 230 132 59 18 37,780
39.94% 21.02% 14.51% 13.76% 6.36% 1.38% 1.05% 0.77% 0.61% 0.35% 0.16% 0.05% 92.56%
$21$ 837 246 21 5 2 1 1,112
75.27% 22.12% 1.89% 0.45% 0.18% 0.09% 91.37%
$22$ 865 117 17 4 6 12 20 13 9 2 1 2 1,068
80.99% 10.96% 1.59% 0.37% 0.56% 1.12% 1.87% 1.22% 0.84% 0.19% 0.09% 0.19% 94.35%
$23$ 181 13 1 195
92.82% 6.67% 0.51% 99.49%
Total 10,161,949 6,217,941 2,366,322 1,767,748 568,047 137,904 24,011 10,097 4,740 1,973 759 227 21,247,651
47.83% 29.26% 11.14% 8.32% 2.67% 0.65% 0.11% 0.05% 0.02% 0.01% 0.00% 0.00% 99.01%

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
25.86% 0.00%
3 159,329 324,863
59.76% 38.50%
4 460,086 373,278 212,634
61.86% 49.86% 12.67%
5 300,892 298,104 563,921
93.40% 93.91% 82.10%
6 470,687 319,380 852,573 1,212,731
53.08% 72.80% 40.22% 43.84%
7 889 23,968 425,305 687,149
93.97% 99.97% 97.49% 97.79%
8 77,348 272,805 282,737 206,966 337,916
68.45% 82.54% 46.50% 38.04% 22.53%
9 5,736 2,762 8,148 61,020 97,478
60.24% 93.34% 91.22% 35.72% 60.81%
10 217,200 15,227 50,916 302,274 727,649 497,983
77.47% 79.91% 57.18% 65.51% 64.01% 59.36%
11 42 6 8 19 609 798
80.77% 100.00% 100.00% 100.00% 95.75% 91.41%
12 1,696 608 3,075 6,204 18,962 21,854 13,948
39.89% 71.87% 66.72% 61.48% 50.23% 42.01% 12.19%
13 28 2 2 2 10 50 174
96.55% 100.00% 100.00% 100.00% 100.00% 98.04% 80.56%
14 189 18 48 47 109 37 484 344
71.05% 100.00% 96.00% 100.00% 89.34% 94.87% 68.75% 26.12%
15 205 13 62 36 76 81 279 394
74.55% 100.00% 91.18% 94.74% 86.36% 86.17% 69.75% 44.93%
16 5,649 109 2,071 461 7,174 988 5,640 1,022 5,851
64.12% 94.78% 77.19% 90.22% 57.57% 68.14% 44.77% 45.50% 12.69%
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 2,152 83 570 502 1,065 431 5,083 592 2,152 3,859
47.57% 92.22% 73.45% 85.81% 74.58% 86.72% 8.03% 79.14% 7.43% 2.39%
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 91 873 249 2,289 882 2,507 752 2,936 818 2,352
62.56% 91.92% 87.56% 95.04% 84.34% 93.14% 81.71% 88.06% 57.28% 80.04% 11.45%
21 139 19 29 42 37 26 86 58 40 192 169
65.88% 100.00% 80.56% 85.71% 77.08% 100.00% 90.53% 73.42% 80.00% 63.58% 85.79%
22 99 13 31 25 91 67 118 46 86 26 161 102
85.34% 92.86% 88.57% 96.15% 87.50% 98.53% 84.89% 95.83% 87.76% 89.66% 86.56% 49.76%
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%