Summary Tables for 5-adic Fields of Degree 15
By Discriminant and Ramification Index
| e=1 | e=3 | e=5 | e=15 | Total
|
---|
d=0 | 1 | | | | 1
|
---|
10 | | 1 | | | 1
|
---|
15 | | | 44 | 4 | 48
|
---|
16 | | | | 4 | 4
|
---|
17 | | | | 4 | 4
|
---|
18 | | | 44 | 8 | 52
|
---|
20 | | | | 20 | 20
|
---|
21 | | | 44 | 20 | 64
|
---|
22 | | | | 40 | 40
|
---|
23 | | | | 20 | 20
|
---|
24 | | | 88 | | 88
|
---|
25 | | | | 100 | 100
|
---|
26 | | | | 200 | 200
|
---|
27 | | | 45 | 100 | 145
|
---|
28 | | | | 100 | 100
|
---|
29 | | | | 125 | 125
|
---|
Total | 1 | 1 | 265 | 745 | 1012
|
---|
By Galois Group and Ramification Index
| e=1 | e=3 | e=5 | e=15 | Total
|
---|
g=1 | 1 | | 5 | | 6
|
---|
2 | | | | 1 | 1
|
---|
3 | | | 3 | | 3
|
---|
4 | | 1 | | 5 | 6
|
---|
6 | | | | 4 | 4
|
---|
7 | | | | 2 | 2
|
---|
8 | | | 17 | | 17
|
---|
9 | | | 2 | | 2
|
---|
11 | | | | 13 | 13
|
---|
12 | | | 6 | | 6
|
---|
13 | | | | 6 | 6
|
---|
14 | | | | 6 | 6
|
---|
17 | | | | 24 | 24
|
---|
18 | | | | 12 | 12
|
---|
19 | | | 24 | | 24
|
---|
25 | | | 48 | | 48
|
---|
27 | | | | 48 | 48
|
---|
30 | | | 24 | | 24
|
---|
31 | | | | 24 | 24
|
---|
32 | | | | 144 | 144
|
---|
37 | | | | 96 | 96
|
---|
38 | | | 136 | | 136
|
---|
40 | | | | 48 | 48
|
---|
49 | | | | 312 | 312
|
---|
Total | 1 | 1 | 265 | 745 | 1012
|
---|
By Galois Group and Discriminant
Ramification Index = 1
| d=0 | Total
|
---|
g=1 | 1 | 1
|
---|
Total | 1 | 1
|
---|
Ramification Index = 3
| d=10 | Total
|
---|
g=4 | 1 | 1
|
---|
Total | 1 | 1
|
---|
Ramification Index = 5
| d=15 | 18 | 21 | 24 | 27 | Total
|
---|
g=1 | | | | 5 | | 5
|
---|
3 | | 2 | | 1 | | 3
|
---|
8 | 4 | 2 | 4 | 2 | 5 | 17
|
---|
9 | | | | 2 | | 2
|
---|
12 | | 4 | | 2 | | 6
|
---|
19 | 8 | 4 | 8 | 4 | | 24
|
---|
25 | | | | 48 | | 48
|
---|
30 | | 16 | | 8 | | 24
|
---|
38 | 32 | 16 | 32 | 16 | 40 | 136
|
---|
Total | 44 | 44 | 44 | 88 | 45 | 265
|
---|
Ramification Index = 15
| d=15 | 16 | 17 | 18 | 20 | 21 | 22 | 23 | 25 | 26 | 27 | 28 | 29 | Total
|
---|
g=2 | | | | | | | | | | 1 | | | | 1
|
---|
4 | | | | | | | | | | 5 | | | | 5
|
---|
6 | | | | | 2 | | | | | 2 | | | | 4
|
---|
7 | | | | | 2 | | | | | | | | | 2
|
---|
11 | | | 4 | | | | | 4 | | | | | 5 | 13
|
---|
13 | | | | 1 | | | 5 | | | | | | | 6
|
---|
14 | | | | 1 | | | 5 | | | | | | | 6
|
---|
17 | | 2 | | 2 | | | 10 | | | | | 10 | | 24
|
---|
18 | | 2 | | | | | | | | | | 10 | | 12
|
---|
27 | 4 | | | | | 4 | | | 20 | | 20 | | | 48
|
---|
31 | | | | | | | | | | 24 | | | | 24
|
---|
32 | | | | 4 | | | 20 | | | 120 | | | | 144
|
---|
37 | | | | | 8 | | | | | 48 | | 40 | | 96
|
---|
40 | | | | | 8 | | | | | | | 40 | | 48
|
---|
49 | | | | | | 16 | | 16 | 80 | | 80 | | 120 | 312
|
---|
Total | 4 | 4 | 4 | 8 | 20 | 20 | 40 | 20 | 100 | 200 | 100 | 100 | 125 | 745
|
---|