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The database currently contains 21,582 $p$-adic fields, including all with $p < 200$ and degree $n < 16$.

Distribution of degree and ramification index for 2-adic fields

ramification index
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Total
degree
1 1 1
100.00% 0.01%
2 1 6 7
14.29% 85.71% 0.09%
3 1 1 2
50.00% 50.00% 0.02%
4 1 10 48 59
1.69% 16.95% 81.36% 0.72%
5 1 1 2
50.00% 50.00% 0.02%
6 1 14 2 30 47
2.13% 29.79% 4.26% 63.83% 0.57%
7 1 1 2
50.00% 50.00% 0.02%
8 1 22 256 1544 1823
0.05% 1.21% 14.04% 84.70% 22.25%
9 1 1 1 3
33.33% 33.33% 33.33% 0.04%
10 1 30 1 126 158
0.63% 18.99% 0.63% 79.75% 1.93%
11 1 1 2
50.00% 50.00% 0.02%
12 1 54 2 1344 156 3936 5493
0.02% 0.98% 0.04% 24.47% 2.84% 71.65% 67.03%
13 1 1 2
50.00% 50.00% 0.02%
14 1 78 1 510 590
0.17% 13.22% 0.17% 86.44% 7.20%
15 1 1 1 1 4
25.00% 25.00% 25.00% 25.00% 0.05%

Distribution of degree and ramification index for 3-adic fields

ramification index
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Total
degree
1 1 1
100.00% 0.03%
2 1 2 3
33.33% 66.67% 0.10%
3 1 9 10
10.00% 90.00% 0.35%
4 1 2 2 5
20.00% 40.00% 40.00% 0.17%
5 1 1 2
50.00% 50.00% 0.07%
6 1 2 21 51 75
1.33% 2.67% 28.00% 68.00% 2.61%
7 1 1 2
50.00% 50.00% 0.07%
8 1 2 3 2 8
12.50% 25.00% 37.50% 25.00% 0.28%
9 1 41 753 795
0.13% 5.16% 94.72% 27.66%
10 1 2 1 2 6
16.67% 33.33% 16.67% 33.33% 0.21%
11 1 1 2
50.00% 50.00% 0.07%
12 1 2 93 2 243 444 785
0.13% 0.25% 11.85% 0.25% 30.96% 56.56% 27.31%
13 1 1 2
50.00% 50.00% 0.07%
14 1 2 1 2 6
16.67% 33.33% 16.67% 33.33% 0.21%
15 1 201 1 969 1172
0.09% 17.15% 0.09% 82.68% 40.78%

Distribution of degree and discriminant exponent for 2-adic fields

discriminant exponent
0 2 3 4 6 8 9 10 11 12 14 15 16 17 18 19 20 21 22 24 25 26 27 28 29 30 31 32 33 34 35 Total
degree
1 1 1
2.17% 0.01%
2 1 2 4 7
0.70% 1.41% 2.82% 0.09%
3 1 1 2
0.70% 0.70% 0.02%
4 1 5 9 8 8 8 20 59
0.31% 1.53% 2.76% 2.45% 2.45% 2.45% 6.13% 0.72%
5 1 1 2
0.64% 0.64% 0.02%
6 1 2 8 4 8 8 16 47
0.19% 0.37% 1.50% 0.75% 1.50% 1.50% 2.99% 0.57%
7 1 1 2
0.57% 0.57% 0.02%
8 1 13 3 30 14 86 32 82 120 64 146 152 128 128 144 128 128 128 296 1823
0.04% 0.57% 0.13% 1.32% 0.61% 3.78% 1.40% 3.60% 5.27% 2.81% 6.41% 6.67% 5.62% 5.62% 6.32% 5.62% 5.62% 5.62% 12.99% 22.25%
9 1 1 1 3
0.09% 0.09% 0.09% 0.04%
10 1 1 16 4 8 16 16 32 64 158
0.12% 0.12% 1.95% 0.49% 0.98% 1.95% 1.95% 3.90% 7.80% 1.93%
11 1 1 2
0.40% 0.40% 0.02%
12 1 2 34 3 28 81 80 144 456 128 128 304 256 256 560 544 888 512 1088 5493
0.01% 0.03% 0.48% 0.04% 0.39% 1.14% 1.12% 2.02% 6.41% 1.80% 1.80% 4.27% 3.60% 3.60% 7.87% 7.64% 12.48% 7.19% 15.29% 67.03%
13 1 1 2
0.32% 0.32% 0.02%
14 1 1 40 4 8 16 40 32 64 128 256 590
0.06% 0.06% 2.44% 0.24% 0.49% 0.98% 2.44% 1.95% 3.91% 7.82% 15.64% 7.20%
15 1 1 1 1 4
0.04% 0.04% 0.04% 0.04% 0.05%

Distribution of degree and discriminant exponent for 3-adic fields

discriminant exponent
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 Total
degree
1 1 1
2.17% 0.03%
2 1 2 3
0.70% 1.41% 0.10%
3 1 2 4 3 10
0.70% 1.41% 2.82% 2.11% 0.35%
4 1 2 2 5
0.31% 0.61% 0.61% 0.17%
5 1 1 2
0.64% 0.64% 0.07%
6 1 2 7 6 10 16 12 21 75
0.19% 0.37% 1.31% 1.12% 1.87% 2.99% 2.24% 3.93% 2.61%
7 1 1 2
0.57% 0.57% 0.07%
8 1 2 3 2 8
0.04% 0.09% 0.13% 0.09% 0.28%
9 1 12 2 26 10 41 22 66 96 54 72 96 54 54 108 81 795
0.09% 1.13% 0.19% 2.45% 0.94% 3.86% 2.07% 6.21% 9.04% 5.08% 6.78% 9.04% 5.08% 5.08% 10.17% 7.63% 27.66%
10 1 2 1 2 6
0.12% 0.24% 0.12% 0.24% 0.21%
11 1 1 2
0.40% 0.40% 0.07%
12 1 2 2 30 6 15 16 52 106 48 63 114 150 180 785
0.01% 0.03% 0.03% 0.42% 0.08% 0.21% 0.22% 0.73% 1.49% 0.67% 0.89% 1.60% 2.11% 2.53% 27.31%
13 1 1 2
0.32% 0.32% 0.07%
14 1 2 1 2 6
0.06% 0.12% 0.06% 0.12% 0.21%
15 1 1 52 4 12 6 100 18 36 108 105 162 324 243 1172
0.04% 0.04% 2.03% 0.16% 0.47% 0.23% 3.91% 0.70% 1.41% 4.23% 4.11% 6.34% 12.68% 9.51% 40.78%

Distribution of Galois groups for 2-adic fields of degree 4

Galois group $C_4$ (as 4T1) $C_2^2$ (as 4T2) $D_4$ (as 4T3) $A_4$ (as 4T4) $S_4$ (as 4T5)
count 12 7 36 1 3
proportion 20.34% 11.86% 61.02% 1.69% 5.08%

Distribution of Galois groups for 2-adic fields of degree 6

Galois group $C_6$ (as 6T1) $S_3$ (as 6T2) $D_6$ (as 6T3) $A_4$ (as 6T4) $C_3\times S_3$ (as 6T5) $C_2\times A_4$ (as 6T6) $S_4$ (as 6T7) $S_4$ (as 6T8) $C_2\times S_4$ (as 6T11)
count 7 1 6 1 1 7 3 3 18
proportion 14.89% 2.13% 12.77% 2.13% 2.13% 14.89% 6.38% 6.38% 38.30%

Distribution of Galois groups for 2-adic fields of degree 8

Galois group $C_8$ (as 8T1) $C_2\times C_4$ (as 8T2) $C_2^3$ (as 8T3) $D_4$ (as 8T4) $Q_8$ (as 8T5) $D_8$ (as 8T6) $OD_{16}$ (as 8T7) $SD_{16}$ (as 8T8) $C_2\times D_4$ (as 8T9) $C_2^2:C_4$ (as 8T10)
count 24 18 1 18 6 32 36 36 36 24
proportion 1.32% 0.99% 0.05% 0.99% 0.33% 1.76% 1.97% 1.97% 1.97% 1.32%
Galois group $D_4:C_2$ (as 8T11) $C_2\times A_4$ (as 8T13) $S_4$ (as 8T14) $C_8:C_2^2$ (as 8T15) $C_2^3.C_4$ (as 8T16) $C_4\wr C_2$ (as 8T17) $C_2^2\wr C_2$ (as 8T18) $C_2^2.D_4$ (as 8T19) $C_2^2.D_4$ (as 8T20) $C_2^2.D_4$ (as 8T21)
count 48 7 3 76 24 96 32 48 24 24
proportion 2.63% 0.38% 0.16% 4.17% 1.32% 5.27% 1.76% 2.63% 1.32% 1.32%
Galois group $\GL(2,3)$ (as 8T23) $C_2\times S_4$ (as 8T24) $F_8$ (as 8T25) $D_4:D_4$ (as 8T26) $C_2\wr C_4$ (as 8T27) $C_2\wr C_4$ (as 8T28) $C_2\wr C_2^2$ (as 8T29) $C_4^2:C_4$ (as 8T30) $C_2\wr C_2^2$ (as 8T31) $C_2^2\wr C_2:C_3$ (as 8T33)
count 16 18 2 96 96 96 96 96 32 14
proportion 0.88% 0.99% 0.11% 5.27% 5.27% 5.27% 5.27% 5.27% 1.76% 0.77%
Galois group $C_2^2:S_4$ (as 8T34) $D_4^2.C_2$ (as 8T35) $F_8:C_3$ (as 8T36) $C_2\wr A_4$ (as 8T38) $C_2^3:A_4:C_2$ (as 8T40) $C_2^2:S_4:C_2$ (as 8T41) $A_4\wr C_2$ (as 8T42) $C_2^3:S_4.C_2$ (as 8T44)
count 1 384 14 56 8 36 5 144
proportion 0.05% 21.06% 0.77% 3.07% 0.44% 1.97% 0.27% 7.90%

Distribution of Galois groups for 2-adic fields of degree 10

Galois group $C_{10}$ (as 10T1) $F_5$ (as 10T4) $C_2\times F_5$ (as 10T5) $C_2^4:C_5$ (as 10T8) $C_2\times C_2^4:C_5$ (as 10T14) $C_2^4:C_5:C_4$ (as 10T24) $C_2^4:C_5:C_4$ (as 10T25) $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29)
count 7 1 6 3 21 15 15 90
proportion 4.43% 0.63% 3.80% 1.90% 13.29% 9.49% 9.49% 56.96%

Distribution of Galois groups for 2-adic fields of degree 12

Galois group $C_{12}$ (as 12T1) $C_2\times C_6$ (as 12T2) $D_6$ (as 12T3) $A_4$ (as 12T4) $C_3:C_4$ (as 12T5) $C_2\times A_4$ (as 12T6) $C_2\times A_4$ (as 12T7) $S_4$ (as 12T8) $S_4$ (as 12T9) $C_2^2\times S_3$ (as 12T10)
count 12 7 3 1 4 7 7 3 3 4
proportion 0.22% 0.13% 0.05% 0.02% 0.07% 0.13% 0.13% 0.05% 0.05% 0.07%
Galois group $C_4\times S_3$ (as 12T11) $D_{12}$ (as 12T12) $C_3:D_4$ (as 12T13) $C_3\times D_4$ (as 12T14) $C_3:D_4$ (as 12T15) $C_6\times S_3$ (as 12T18) $C_3\times C_3:C_4$ (as 12T19) $C_2\times S_4$ (as 12T21) $C_2\times S_4$ (as 12T22) $C_2\times S_4$ (as 12T23)
count 8 4 4 36 4 3 4 9 9 18
proportion 0.15% 0.07% 0.07% 0.66% 0.07% 0.05% 0.07% 0.16% 0.16% 0.33%
Galois group $C_2\times S_4$ (as 12T24) $C_2^2\times A_4$ (as 12T25) $C_2^2\times A_4$ (as 12T26) $A_4:C_4$ (as 12T27) $S_3\times D_4$ (as 12T28) $C_4\times A_4$ (as 12T29) $A_4:C_4$ (as 12T30) $C_4^2:C_3$ (as 12T31) $C_3\times C_3:D_4$ (as 12T42) $S_3\times A_4$ (as 12T43)
count 18 21 14 12 24 12 12 2 8 1
proportion 0.33% 0.38% 0.25% 0.22% 0.44% 0.22% 0.22% 0.04% 0.15% 0.02%
Galois group $C_3\times S_4$ (as 12T45) $C_2^2\times S_4$ (as 12T48) $\GL(2,Z/4)$ (as 12T49) $\GL(2,Z/4)$ (as 12T50) $D_4\times A_4$ (as 12T51) $\GL(2,Z/4)$ (as 12T52) $C_4\times S_4$ (as 12T53) $C_4:S_4$ (as 12T54) $C_2\times C_4^2:C_3$ (as 12T55) $C_2^2\wr C_2:C_3$ (as 12T58)
count 3 36 24 12 36 12 24 12 14 14
proportion 0.05% 0.66% 0.44% 0.22% 0.66% 0.22% 0.44% 0.22% 0.25% 0.25%
Galois group $C_2^2\wr C_2:C_3$ (as 12T59) $C_4:D_4:C_3$ (as 12T60) $C_4:D_4:C_3$ (as 12T61) $C_4^2:C_3:C_2$ (as 12T62) $C_4^2:C_3:C_2$ (as 12T63) $C_4^2:C_3:C_2$ (as 12T64) $C_4^2:C_3:C_2$ (as 12T65) $C_2^2:S_4$ (as 12T66) $C_2^2:S_4$ (as 12T67) $C_2^2:S_4$ (as 12T68)
count 14 14 14 6 6 6 6 3 1 3
proportion 0.25% 0.25% 0.25% 0.11% 0.11% 0.11% 0.11% 0.05% 0.02% 0.05%
Galois group $C_2^2:S_4$ (as 12T69) $D_4\times S_4$ (as 12T86) $C_2\times C_2^2\wr C_2:C_3$ (as 12T87) $C_2\times C_2^2\wr C_2:C_3$ (as 12T88) $C_2\times C_4:D_4:C_3$ (as 12T89) $C_2\times C_4:D_4:C_3$ (as 12T92) $C_4\times C_4^2:C_3$ (as 12T94) $C_2\times C_4^2:C_3:C_2$ (as 12T95) $C_2\times C_4^2:C_3:C_2$ (as 12T96) $C_2\times C_4^2:C_3:C_2$ (as 12T97)
count 1 72 42 42 42 42 48 36 18 18
proportion 0.02% 1.31% 0.76% 0.76% 0.76% 0.76% 0.87% 0.66% 0.33% 0.33%
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Galois group $C_{12}$ (as 12T1) $C_2\times C_6$ (as 12T2) $D_6$ (as 12T3) $A_4$ (as 12T4) $C_3:C_4$ (as 12T5) $C_2\times A_4$ (as 12T6) $C_2\times A_4$ (as 12T7) $S_4$ (as 12T8) $S_4$ (as 12T9) $C_2^2\times S_3$ (as 12T10)
count 12 7 3 1 4 7 7 3 3 4
proportion 0.22% 0.13% 0.05% 0.02% 0.07% 0.13% 0.13% 0.05% 0.05% 0.07%
Galois group $C_4\times S_3$ (as 12T11) $D_{12}$ (as 12T12) $C_3:D_4$ (as 12T13) $C_3\times D_4$ (as 12T14) $C_3:D_4$ (as 12T15) $C_6\times S_3$ (as 12T18) $C_3\times C_3:C_4$ (as 12T19) $C_2\times S_4$ (as 12T21) $C_2\times S_4$ (as 12T22) $C_2\times S_4$ (as 12T23)
count 8 4 4 36 4 3 4 9 9 18
proportion 0.15% 0.07% 0.07% 0.66% 0.07% 0.05% 0.07% 0.16% 0.16% 0.33%
Galois group $C_2\times S_4$ (as 12T24) $C_2^2\times A_4$ (as 12T25) $C_2^2\times A_4$ (as 12T26) $A_4:C_4$ (as 12T27) $S_3\times D_4$ (as 12T28) $C_4\times A_4$ (as 12T29) $A_4:C_4$ (as 12T30) $C_4^2:C_3$ (as 12T31) $C_3\times C_3:D_4$ (as 12T42) $S_3\times A_4$ (as 12T43)
count 18 21 14 12 24 12 12 2 8 1
proportion 0.33% 0.38% 0.25% 0.22% 0.44% 0.22% 0.22% 0.04% 0.15% 0.02%
Galois group $C_3\times S_4$ (as 12T45) $C_2^2\times S_4$ (as 12T48) $\GL(2,Z/4)$ (as 12T49) $\GL(2,Z/4)$ (as 12T50) $D_4\times A_4$ (as 12T51) $\GL(2,Z/4)$ (as 12T52) $C_4\times S_4$ (as 12T53) $C_4:S_4$ (as 12T54) $C_2\times C_4^2:C_3$ (as 12T55) $C_2^2\wr C_2:C_3$ (as 12T58)
count 3 36 24 12 36 12 24 12 14 14
proportion 0.05% 0.66% 0.44% 0.22% 0.66% 0.22% 0.44% 0.22% 0.25% 0.25%
Galois group $C_2^2\wr C_2:C_3$ (as 12T59) $C_4:D_4:C_3$ (as 12T60) $C_4:D_4:C_3$ (as 12T61) $C_4^2:C_3:C_2$ (as 12T62) $C_4^2:C_3:C_2$ (as 12T63) $C_4^2:C_3:C_2$ (as 12T64) $C_4^2:C_3:C_2$ (as 12T65) $C_2^2:S_4$ (as 12T66) $C_2^2:S_4$ (as 12T67) $C_2^2:S_4$ (as 12T68)
count 14 14 14 6 6 6 6 3 1 3
proportion 0.25% 0.25% 0.25% 0.11% 0.11% 0.11% 0.11% 0.05% 0.02% 0.05%
Galois group $C_2^2:S_4$ (as 12T69) $D_4\times S_4$ (as 12T86) $C_2\times C_2^2\wr C_2:C_3$ (as 12T87) $C_2\times C_2^2\wr C_2:C_3$ (as 12T88) $C_2\times C_4:D_4:C_3$ (as 12T89) $C_2\times C_4:D_4:C_3$ (as 12T92) $C_4\times C_4^2:C_3$ (as 12T94) $C_2\times C_4^2:C_3:C_2$ (as 12T95) $C_2\times C_4^2:C_3:C_2$ (as 12T96) $C_2\times C_4^2:C_3:C_2$ (as 12T97)
count 1 72 42 42 42 42 48 36 18 18
proportion 0.02% 1.31% 0.76% 0.76% 0.76% 0.76% 0.87% 0.66% 0.33% 0.33%
Galois group $C_2^2.A_4:C_4$ (as 12T98) $C_2^5.C_6$ (as 12T99) $C_2\times C_2^2:S_4$ (as 12T100) $C_2\times C_2^2:S_4$ (as 12T101) $C_2^2:(A_4:C_4)$ (as 12T102) $C_2\times C_2^2:S_4$ (as 12T103) $(C_2^2.\SL(2,3)):C_2$ (as 12T104) $C_2^5.C_6$ (as 12T105) $C_2\times C_2^2:S_4$ (as 12T106) $C_2^2:(A_4:C_4)$ (as 12T107)
count 48 24 9 18 12 18 64 24 3 4
proportion 0.87% 0.44% 0.16% 0.33% 0.22% 0.33% 1.17% 0.44% 0.05% 0.07%
Galois group $C_2^2:S_4:C_2$ (as 12T108) $C_2^2:S_4:C_2$ (as 12T109) $C_2^2:S_4:C_2$ (as 12T110) $C_2^2:S_4:C_2$ (as 12T111) $C_4^2:C_3:C_2^2$ (as 12T112) $C_4^2:C_3:C_2^2$ (as 12T113) $C_4^2:C_3:C_2^2$ (as 12T114) $C_4^2:C_3:C_2^2$ (as 12T115) $A_4\wr C_2$ (as 12T126) $A_4\wr C_2$ (as 12T128)
count 36 36 36 36 36 36 36 36 5 5
proportion 0.66% 0.66% 0.66% 0.66% 0.66% 0.66% 0.66% 0.66% 0.09% 0.09%
Galois group $A_4\wr C_2$ (as 12T129) $C_2^5.(C_2\times C_6)$ (as 12T134) $C_2^4:C_3.D_4$ (as 12T135) $C_2\times C_2^2:S_4:C_2$ (as 12T136) $C_2\times C_2^2:S_4:C_2$ (as 12T137) $C_2\times C_4^2:C_3:C_2^2$ (as 12T138) $C_2^2\times C_2^2:S_4$ (as 12T139) $C_2\times C_4^2:C_3:C_2^2$ (as 12T140) $C_4^2.(C_3\times D_4)$ (as 12T141) $C_2^5.(C_2\times C_6)$ (as 12T142)
count 5 144 8 72 72 72 24 72 144 144
proportion 0.09% 2.62% 0.15% 1.31% 1.31% 1.31% 0.44% 1.31% 2.62% 2.62%
Galois group $C_2^2.(D_4.A_4)$ (as 12T143) $C_2^4:C_3.D_4$ (as 12T145) $C_2^4:C_3.D_4$ (as 12T146) $C_4^2:C_3.D_4$ (as 12T147) $C_2^4:C_3.D_4$ (as 12T148) $C_4^2:C_3.D_4$ (as 12T149) $C_4\times C_4^2:C_3:C_2$ (as 12T150) $C_4^2.D_{12}$ (as 12T151) $C_2^4.D_{12}$ (as 12T152) $C_2^4.(C_4\times S_3)$ (as 12T153)
count 384 48 48 48 24 48 96 48 48 48
proportion 6.99% 0.87% 0.87% 0.87% 0.44% 0.87% 1.75% 0.87% 0.87% 0.87%
Galois group $C_2^4.D_{12}$ (as 12T154) $C_2^4.(C_4\times S_3)$ (as 12T155) 12T158 12T159 12T166 12T185 12T186 12T193 12T205 12T206
count 48 48 15 20 7 288 288 288 15 15
proportion 0.87% 0.87% 0.27% 0.36% 0.13% 5.24% 5.24% 5.24% 0.27% 0.27%
Galois group 12T208 12T221 12T223 12T224 12T225 12T250 12T254
count 40 192 192 192 192 576 63
proportion 0.73% 3.50% 3.50% 3.50% 3.50% 10.49% 1.15%
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Distribution of Galois groups for 2-adic fields of degree 14

Galois group $C_{14}$ (as 14T1) $C_2\times C_7:C_3$ (as 14T5) $F_8$ (as 14T6) $C_2\times F_8$ (as 14T9) $F_8:C_3$ (as 14T11) $C_2\times F_8:C_3$ (as 14T18) $C_2^3:F_8$ (as 14T21) $C_2 \wr C_7$ (as 14T29) 14T35 14T44
count 7 7 2 14 14 98 7 49 49 343
proportion 1.19% 1.19% 0.34% 2.37% 2.37% 16.61% 1.19% 8.31% 8.31% 58.14%

Distribution of Galois groups for 3-adic fields of degree 6

Galois group $C_6$ (as 6T1) $S_3$ (as 6T2) $D_6$ (as 6T3) $C_3\times S_3$ (as 6T5) $S_3^2$ (as 6T9) $C_3:S_3.C_2$ (as 6T10) $S_3\wr C_2$ (as 6T13)
count 12 6 12 24 9 4 8
proportion 16.00% 8.00% 16.00% 32.00% 12.00% 5.33% 10.67%

Distribution of Galois groups for 3-adic fields of degree 9

Galois group $C_9$ (as 9T1) $C_3^2$ (as 9T2) $D_9$ (as 9T3) $C_3\times S_3$ (as 9T4) $C_3:S_3$ (as 9T5) $C_9:C_3$ (as 9T6) $He_3$ (as 9T7) $S_3^2$ (as 9T8) $C_3:S_3.C_2$ (as 9T9) $D_9:C_3$ (as 9T10)
count 12 1 5 24 1 8 4 9 2 49
proportion 1.51% 0.13% 0.63% 3.02% 0.13% 1.01% 0.50% 1.13% 0.25% 6.16%
Galois group $He_3:C_2$ (as 9T11) $C_3^2:S_3$ (as 9T12) $He_3:C_2$ (as 9T13) $PSU(3,2)$ (as 9T14) $F_9$ (as 9T15) $S_3\wr C_2$ (as 9T16) $C_3\wr C_3$ (as 9T17) $C_3.S_3^2$ (as 9T18) $PSU(3,2):C_2$ (as 9T19) $C_3\wr S_3$ (as 9T20)
count 20 36 20 4 4 4 36 48 16 180
proportion 2.52% 4.53% 2.52% 0.50% 0.50% 0.50% 4.53% 6.04% 2.01% 22.64%
Galois group $C_3\wr C_3:C_2$ (as 9T21) $C_3^3:C_6$ (as 9T22) $C_3\wr S_3:C_2$ (as 9T24)
count 108 60 144
proportion 13.58% 7.55% 18.11%

Distribution of Galois groups for 3-adic fields of degree 12

Galois group $C_{12}$ (as 12T1) $C_2\times C_6$ (as 12T2) $D_6$ (as 12T3) $C_3:C_4$ (as 12T5) $C_4\times S_3$ (as 12T11) $D_{12}$ (as 12T12) $C_3:D_4$ (as 12T13) $C_3\times D_4$ (as 12T14) $C_3:D_4$ (as 12T15) $S_3^2$ (as 12T16)
count 8 4 6 2 10 2 5 8 5 9
proportion 1.02% 0.51% 0.76% 0.25% 1.27% 0.25% 0.64% 1.02% 0.64% 1.15%
Galois group $C_3:S_3.C_2$ (as 12T17) $C_6\times S_3$ (as 12T18) $C_3\times C_3:C_4$ (as 12T19) $S_3\wr C_2$ (as 12T34) $S_3\wr C_2$ (as 12T35) $S_3\wr C_2$ (as 12T36) $C_3^2:D_4$ (as 12T38) $C_2.S_3^2$ (as 12T39) $C_2\times C_3:S_3.C_2$ (as 12T40) $C_2\times C_3:S_3.C_2$ (as 12T41)
count 4 24 8 8 8 8 10 8 4 4
proportion 0.51% 3.06% 1.02% 1.02% 1.02% 1.02% 1.27% 1.02% 0.51% 0.51%
Galois group $C_3\times C_3:D_4$ (as 12T42) $F_9$ (as 12T46) $PSU(3,2)$ (as 12T47) $C_3\times S_3^2$ (as 12T70) $C_3^3:C_2^2$ (as 12T71) $C_3^2:(C_3:C_4)$ (as 12T72) $C_3\times C_3:S_3.C_2$ (as 12T73) $PSU(3,2):C_2$ (as 12T84) $S_3^2:S_3$ (as 12T116) $C_3:S_3\wr C_2$ (as 12T118)
count 40 4 4 36 4 4 16 16 20 8
proportion 5.10% 0.51% 0.51% 4.59% 0.51% 0.51% 2.04% 2.04% 2.55% 1.02%
Galois group $S_3\times C_3:S_3.C_2$ (as 12T119) $S_3^2:S_3$ (as 12T120) $C_3\times S_3\wr C_2$ (as 12T121) $C_3\times C_3^3:C_2^2$ (as 12T130) $C_3\times C_3^2:(C_3:C_4)$ (as 12T131) 12T167 12T169 12T170 12T171 12T172
count 20 20 32 32 32 160 40 32 8 6
proportion 2.55% 2.55% 4.08% 4.08% 4.08% 20.38% 5.10% 4.08% 1.02% 0.76%
Galois group 12T173 12T174 12T212 12T215
count 16 6 64 20
proportion 2.04% 0.76% 8.15% 2.55%

Distribution of Galois groups for 3-adic fields of degree 15

Galois group $C_{15}$ (as 15T1) $C_5\times S_3$ (as 15T4) $C_3:F_5$ (as 15T6) $C_3\times F_5$ (as 15T8) $S_3\times F_5$ (as 15T11) $C_3^4:C_5$ (as 15T26) 15T33 15T36 15T41 15T42
count 4 6 1 4 5 8 24 64 40 40
proportion 0.34% 0.51% 0.09% 0.34% 0.43% 0.68% 2.05% 5.46% 3.41% 3.41%
Galois group 15T44 15T52 15T54 15T56 15T64
count 96 80 80 320 400
proportion 8.19% 6.83% 6.83% 27.30% 34.13%

Distribution of Galois groups for 5-adic fields of degree 10

Galois group $C_{10}$ (as 10T1) $D_5$ (as 10T2) $D_{10}$ (as 10T3) $F_5$ (as 10T4) $C_2\times F_5$ (as 10T5) $C_5\times D_5$ (as 10T6) $D_5^2$ (as 10T9) $C_5:F_5$ (as 10T10) $D_5:F_5$ (as 10T17) $C_5:D_5.C_4$ (as 10T18)
count 18 3 6 17 34 36 6 22 44 24
proportion 6.98% 1.16% 2.33% 6.59% 13.18% 13.95% 2.33% 8.53% 17.05% 9.30%
Galois group $D_5^2.C_4$ (as 10T28)
count 48
proportion 18.60%

Distribution of Galois groups for 5-adic fields of degree 15

Galois group $C_{15}$ (as 15T1) $D_{15}$ (as 15T2) $C_3\times D_5$ (as 15T3) $C_5\times S_3$ (as 15T4) $C_3:F_5$ (as 15T6) $S_3\times D_5$ (as 15T7) $C_3\times F_5$ (as 15T8) $C_5^2:C_3$ (as 15T9) $S_3\times F_5$ (as 15T11) $C_5^2:C_6$ (as 15T12)
count 6 1 3 6 4 2 17 2 13 6
proportion 0.59% 0.10% 0.30% 0.59% 0.40% 0.20% 1.68% 0.20% 1.28% 0.59%
Galois group $C_5^2:S_3$ (as 15T13) $C_5^2:S_3$ (as 15T14) $C_5^2:(C_3:C_4)$ (as 15T17) $C_5^2:D_6$ (as 15T18) $C_5^2:C_{12}$ (as 15T19) $C_5\times C_5^2:C_3$ (as 15T25) 15T27 15T30 15T31 15T32
count 6 6 24 12 24 48 48 24 24 144
proportion 0.59% 0.59% 2.37% 1.19% 2.37% 4.74% 4.74% 2.37% 2.37% 14.23%
Galois group 15T37 15T38 15T40 15T49
count 96 136 48 312
proportion 9.49% 13.44% 4.74% 30.83%

Distribution of Galois groups for 7-adic fields of degree 14

Galois group $C_{14}$ (as 14T1) $D_7$ (as 14T2) $D_{14}$ (as 14T3) $F_7$ (as 14T4) $C_2\times C_7:C_3$ (as 14T5) $C_2\times F_7$ (as 14T7) $C_7\times D_7$ (as 14T8) $C_7:D_7.C_2$ (as 14T12) $D_7^2$ (as 14T13) $C_7^2:C_6$ (as 14T14)
count 24 3 6 31 24 62 72 8 9 93
proportion 3.67% 0.46% 0.92% 4.74% 3.67% 9.48% 11.01% 1.22% 1.38% 14.22%
Galois group $D_7\wr C_2$ (as 14T20) 14T23 14T24 14T32
count 16 64 114 128
proportion 2.45% 9.79% 17.43% 19.57%