Summary Tables for 7-adic Fields of Degree 14
By Discriminant and Ramification Index
| e=1 | e=2 | e=7 | e=14 | Total
|
---|
d=0 | 1 | | | | 1
|
---|
7 | | 2 | | | 2
|
---|
14 | | | 27 | 6 | 33
|
---|
15 | | | | 12 | 12
|
---|
16 | | | 27 | 6 | 33
|
---|
17 | | | | 12 | 12
|
---|
18 | | | 27 | 6 | 33
|
---|
19 | | | | 18 | 18
|
---|
20 | | | 27 | | 27
|
---|
21 | | | | 48 | 48
|
---|
22 | | | 27 | 42 | 69
|
---|
23 | | | | 48 | 48
|
---|
24 | | | 54 | 42 | 96
|
---|
25 | | | | 96 | 96
|
---|
26 | | | 28 | 42 | 70
|
---|
27 | | | | 56 | 56
|
---|
Total | 1 | 2 | 217 | 434 | 654
|
---|
By Galois Group and Ramification Index
| e=1 | e=2 | e=7 | e=14 | Total
|
---|
g=1 | 1 | 2 | 7 | 14 | 24
|
---|
2 | | | 1 | 2 | 3
|
---|
3 | | | 2 | 4 | 6
|
---|
4 | | | 8 | 23 | 31
|
---|
5 | | | 8 | 16 | 24
|
---|
7 | | | 23 | 39 | 62
|
---|
8 | | | 24 | 48 | 72
|
---|
12 | | | 8 | | 8
|
---|
13 | | | 3 | 6 | 9
|
---|
14 | | | 24 | 69 | 93
|
---|
20 | | | | 16 | 16
|
---|
23 | | | 64 | | 64
|
---|
24 | | | 45 | 69 | 114
|
---|
32 | | | | 128 | 128
|
---|
Total | 1 | 2 | 217 | 434 | 654
|
---|
By Galois Group and Discriminant
Ramification Index = 1
| d=0 | Total
|
---|
g=1 | 1 | 1
|
---|
Total | 1 | 1
|
---|
Ramification Index = 2
| d=7 | Total
|
---|
g=1 | 2 | 2
|
---|
Total | 2 | 2
|
---|
Ramification Index = 7
| d=14 | 16 | 18 | 20 | 22 | 24 | 26 | Total
|
---|
g=1 | | | | | | 7 | | 7
|
---|
2 | | | | | | 1 | | 1
|
---|
3 | | | 2 | | | | | 2
|
---|
4 | | 3 | | 3 | | 2 | | 8
|
---|
5 | | 3 | | 3 | | 2 | | 8
|
---|
7 | 6 | | 4 | | 6 | | 7 | 23
|
---|
8 | | | | | | 24 | | 24
|
---|
12 | | | 4 | | | 4 | | 8
|
---|
13 | | | 3 | | | | | 3
|
---|
14 | | 9 | | 9 | | 6 | | 24
|
---|
23 | 12 | 12 | 8 | 12 | 12 | 8 | | 64
|
---|
24 | 9 | | 6 | | 9 | | 21 | 45
|
---|
Total | 27 | 27 | 27 | 27 | 27 | 54 | 28 | 217
|
---|
Ramification Index = 14
| d=14 | 15 | 16 | 17 | 18 | 19 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | Total
|
---|
g=1 | | | | | | | | | | | 14 | | | 14
|
---|
2 | | | | | | 2 | | | | | | | | 2
|
---|
3 | | | | | | 2 | | | | | 2 | | | 4
|
---|
4 | | 6 | | | | 4 | | | 6 | | | | 7 | 23
|
---|
5 | | | | 6 | | | 6 | | | | 4 | | | 16
|
---|
7 | | 6 | | 6 | | 4 | 6 | | 6 | | 4 | | 7 | 39
|
---|
8 | | | | | | 6 | | | | | 42 | | | 48
|
---|
13 | | | | | | | | | | | 6 | | | 6
|
---|
14 | | | | | | | 18 | | 18 | | 12 | | 21 | 69
|
---|
20 | | | 2 | | | | | 14 | | | | | | 16
|
---|
24 | | | | | | | 18 | | 18 | | 12 | | 21 | 69
|
---|
32 | 6 | | 4 | | 6 | | | 28 | | 42 | | 42 | | 128
|
---|
Total | 6 | 12 | 6 | 12 | 6 | 18 | 48 | 42 | 48 | 42 | 96 | 42 | 56 | 434
|
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