The database contains $22{,}171{,}096$ 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% 6.18%
3 $266{,}649$ $913{,}287$ $1{,}179{,}936$
22.60% 77.40% 5.32%
4 $743{,}786$ $748{,}622$ $1{,}678{,}348$ $3{,}170{,}756$
23.46% 23.61% 52.93% 14.30%
5 $322{,}140$ $317{,}423$ $686{,}902$ $1{,}326{,}465$
24.29% 23.93% 51.78% 5.98%
6 $886{,}745$ $438{,}679$ $2{,}119{,}888$ $2{,}766{,}495$ $6{,}211{,}807$
14.28% 7.06% 34.13% 44.54% 28.02%
7 $946$ $23{,}975$ $436{,}253$ $702{,}701$ $1{,}163{,}875$
0.08% 2.06% 37.48% 60.38% 5.25%
8 $142{,}583$ $330{,}522$ $608{,}057$ $545{,}668$ $1{,}520{,}570$ $3{,}147{,}400$
4.53% 10.50% 19.32% 17.34% 48.31% 14.20%
9 $9{,}522$ $2{,}959$ $8{,}932$ $171{,}047$ $160{,}633$ $353{,}093$
2.70% 0.84% 2.53% 48.44% 45.49% 1.59%
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% 12.76%
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% 1.01%
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 $8{,}833$ $174$ $2{,}994$ $1{,}757$ $15{,}940$ $8{,}600$ $24{,}645$ $15{,}854$ $81{,}713$ $160{,}510$
5.50% 0.11% 1.87% 1.09% 9.93% 5.36% 15.35% 9.88% 50.91% 0.72%
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% 1.64%
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% 0.18%
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 $3{,}359{,}388$ $3{,}481{,}308$ $5{,}637{,}027$ $4{,}660{,}226$ $2{,}876{,}914$ $905{,}039$ $224{,}824$ $20{,}265$ $121{,}395$ $241{,}449$ $22{,}873$ $370$

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 $21{,}871$ $143{,}933$ $2{,}177{,}637$ $29{,}534$ $797{,}781$ $3{,}170{,}756$
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 $21{,}145$ $136{,}022$ $320{,}818$ $5{,}261$ $22{,}119$ $179{,}812$ $244{,}008$ $46{,}108$ $65{,}311$ $2{,}319$ $923{,}486$ $110{,}157$ $1{,}946{,}133$ $1{,}011{,}145$ $9{,}721$ $1{,}168{,}242$ $6{,}211{,}807$
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% 0.34% 5.05% 0.32% 4.06% 2.30% 5.75% 14.66% 4.59%
$D_n$ $=S_3$ $2{,}177{,}637$ $16{,}306$ $320{,}818$ $2{,}135$ $30{,}398$ $737$ $7{,}457$ $57$ $302$ $78$ $426$ $161$ $105$ $12$ $10$ $11$ $9$ $3$ $8$ $10$
68.68% 1.23% 5.16% 0.18% 0.97% 0.21% 0.26% 3.57% 0.13% 25.00% 16.52% 6.53% 0.07% 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% 0.00% 1.38% 0.00% 0.51% 0.00% 0.16% 0.09% 1.02%
$S_n$ $1{,}175{,}258$ $797{,}781$ $1{,}164{,}813$ $1{,}168{,}242$ $1{,}062{,}232$ $4{,}316$ $3{,}544$ $59{,}915$ $1{,}143$ $162$ $194$ $160$ $173$ $171$ $173$ $165$ $164$ $165$ $168$ $168$ $165$
99.60% 25.16% 87.81% 18.81% 91.27% 0.14% 1.00% 2.12% 71.62% 0.07% 62.18% 6.20% 7.02% 0.11% 79.36% 0.05% 83.25% 0.40% 13.80% 14.84% 84.18%

Distribution by class number

Class numbers have been computed for 96.38% 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
$2$ $177{,}168$ $424{,}929$ $124{,}112$ $585{,}283$ $59{,}025$ $136$ $6$ $0$ $0$ $0$ $0$ $0$ $1{,}370{,}659$
12.93% 31.00% 9.05% 42.70% 4.31% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 100.00%
$3$ $484{,}192$ $389{,}451$ $146{,}404$ $61{,}199$ $21{,}954$ $5{,}992$ $1{,}160$ $154$ $23$ $2$ $0$ $0$ $1{,}110{,}531$
43.60% 35.07% 13.18% 5.51% 1.98% 0.54% 0.10% 0.01% 0.00% 0.00% 0.00% 0.00% 94.12%
$4$ $1{,}046{,}000$ $896{,}785$ $445{,}528$ $427{,}257$ $271{,}007$ $81{,}101$ $2{,}949$ $90$ $1$ $0$ $0$ $0$ $3{,}170{,}718$
32.99% 28.28% 14.05% 13.48% 8.55% 2.56% 0.09% 0.00% 0.00% 0.00% 0.00% 0.00% 100.00%
$5$ $1{,}162{,}917$ $154{,}722$ $8{,}287$ $508$ $24$ $1$ $3$ $2$ $0$ $0$ $0$ $0$ $1{,}326{,}464$
87.67% 11.66% 0.62% 0.04% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 100.00%
$6$ $2{,}855{,}378$ $2{,}102{,}554$ $759{,}790$ $361{,}015$ $117{,}337$ $12{,}319$ $2{,}432$ $643$ $271$ $48$ $6$ $1$ $6{,}211{,}794$
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%
$7$ $1{,}137{,}311$ $26{,}310$ $245$ $7$ $1$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1{,}163{,}874$
97.72% 2.26% 0.02% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 100.00%
$8$ $1{,}178{,}441$ $1{,}092{,}035$ $576{,}764$ $196{,}634$ $55{,}299$ $15{,}673$ $6{,}895$ $5{,}241$ $4{,}157$ $3{,}536$ $4{,}122$ $3{,}112$ $3{,}142{,}554$
37.50% 34.75% 18.35% 6.26% 1.76% 0.50% 0.22% 0.17% 0.13% 0.11% 0.13% 0.10% 99.85%
$9$ $175{,}294$ $103{,}509$ $50{,}462$ $18{,}790$ $3{,}463$ $761$ $217$ $75$ $48$ $19$ $15$ $10$ $352{,}664$
49.71% 29.35% 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%
$11$ $1{,}482$ $87$ $25$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1{,}594$
92.97% 5.46% 1.57% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 99.87%
$12$ $66{,}347$ $105{,}727$ $39{,}708$ $7{,}190$ $2{,}139$ $1{,}685$ $945$ $240$ $17$ $0$ $0$ $0$ $223{,}998$
29.62% 47.20% 17.73% 3.21% 0.95% 0.75% 0.42% 0.11% 0.01% 0.00% 0.00% 0.00% 99.99%
$13$ $268$ $11$ $33$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $312$
85.90% 3.53% 10.58% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 100.00%
$14$ $1{,}276$ $876$ $183$ $53$ $46$ $46$ $46$ $27$ $9$ $1$ $0$ $0$ $2{,}563$
49.79% 34.18% 7.14% 2.07% 1.79% 1.79% 1.79% 1.05% 0.35% 0.04% 0.00% 0.00% 99.38%
$15$ $1{,}146$ $487$ $189$ $28$ $2$ $0$ $0$ $0$ $0$ $1$ $0$ $0$ $1{,}853$
61.85% 26.28% 10.20% 1.51% 0.11% 0.00% 0.00% 0.00% 0.00% 0.05% 0.00% 0.00% 75.17%
$16$ $83{,}976$ $41{,}567$ $10{,}815$ $8{,}831$ $6{,}721$ $3{,}501$ $2{,}151$ $1{,}382$ $707$ $307$ $98$ $24$ $160{,}113$
52.45% 25.96% 6.75% 5.52% 4.20% 2.19% 1.34% 0.86% 0.44% 0.19% 0.06% 0.01% 99.75%
$17$ $199$ $12$ $6$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $217$
91.71% 5.53% 2.76% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 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%
$19$ $185$ $12$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $197$
93.91% 6.09% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 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%
$21$ $837$ $246$ $21$ $5$ $2$ $1$ $0$ $0$ $0$ $0$ $0$ $0$ $1{,}112$
75.27% 22.12% 1.89% 0.45% 0.18% 0.09% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 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%
$23$ $181$ $13$ $1$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $195$
92.82% 6.67% 0.51% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 99.49%
Total $10{,}217{,}788$ $6{,}242{,}647$ $2{,}373{,}647$ $1{,}776{,}648$ $573{,}357$ $139{,}039$ $25{,}346$ $12{,}113$ $7{,}685$ $5{,}203$ $4{,}805$ $3{,}331$ $21{,}368{,}182$
47.82% 29.21% 11.11% 8.31% 2.68% 0.65% 0.12% 0.06% 0.04% 0.02% 0.02% 0.02% 96.38%

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 $460{,}086$ $373{,}279$ $212{,}635$
61.86% 49.86% 12.67%
5 $300{,}892$ $298{,}104$ $563{,}921$
93.40% 93.91% 82.10%
6 $470{,}687$ $319{,}380$ $852{,}576$ $1{,}212{,}735$
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 $77{,}956$ $272{,}805$ $282{,}759$ $207{,}005$ $337{,}916$
54.80% 82.54% 46.50% 38.04% 22.27%
9 $5{,}736$ $2{,}762$ $8{,}148$ $61{,}113$ $97{,}535$
60.24% 93.34% 91.22% 35.76% 60.83%
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 $5{,}650$ $168$ $2{,}380$ $1{,}707$ $10{,}601$ $8{,}123$ $17{,}527$ $14{,}315$ $23{,}505$
64.12% 96.55% 79.55% 97.15% 66.71% 94.46% 71.15% 90.30% 28.88%
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%