Summary Tables for 3-adic Fields of Degree 15
By Discriminant and Ramification Index
| e=1 | e=3 | e=5 | e=15 | Total
|
---|
d=0 | 1 | | | | 1
|
---|
12 | | | 1 | | 1
|
---|
15 | | 50 | | 2 | 52
|
---|
16 | | | | 4 | 4
|
---|
18 | | | | 12 | 12
|
---|
19 | | | | 6 | 6
|
---|
20 | | 100 | | | 100
|
---|
21 | | | | 18 | 18
|
---|
22 | | | | 36 | 36
|
---|
24 | | | | 108 | 108
|
---|
25 | | 51 | | 54 | 105
|
---|
27 | | | | 162 | 162
|
---|
28 | | | | 324 | 324
|
---|
29 | | | | 243 | 243
|
---|
Total | 1 | 201 | 1 | 969 | 1172
|
---|
By Galois Group and Ramification Index
| e=1 | e=3 | e=5 | e=15 | Total
|
---|
g=1 | 1 | 3 | | | 4
|
---|
4 | | 6 | | | 6
|
---|
6 | | | | 1 | 1
|
---|
8 | | | 1 | 3 | 4
|
---|
11 | | | | 5 | 5
|
---|
26 | | 8 | | | 8
|
---|
33 | | 24 | | | 24
|
---|
36 | | 64 | | | 64
|
---|
41 | | | | 40 | 40
|
---|
42 | | | | 40 | 40
|
---|
44 | | 96 | | | 96
|
---|
52 | | | | 80 | 80
|
---|
54 | | | | 80 | 80
|
---|
56 | | | | 320 | 320
|
---|
64 | | | | 400 | 400
|
---|
Total | 1 | 201 | 1 | 969 | 1172
|
---|
By Galois Group and Discriminant
Ramification Index = 1
| d=0 | Total
|
---|
g=1 | 1 | 1
|
---|
Total | 1 | 1
|
---|
Ramification Index = 3
| d=15 | 20 | 25 | Total
|
---|
g=1 | | 3 | | 3
|
---|
4 | 2 | 1 | 3 | 6
|
---|
26 | | 8 | | 8
|
---|
33 | 16 | 8 | | 24
|
---|
36 | | 64 | | 64
|
---|
44 | 32 | 16 | 48 | 96
|
---|
Total | 50 | 100 | 51 | 201
|
---|
Ramification Index = 5
| d=12 | Total
|
---|
g=8 | 1 | 1
|
---|
Total | 1 | 1
|
---|
Ramification Index = 15
| d=15 | 16 | 18 | 19 | 21 | 22 | 24 | 25 | 27 | 28 | 29 | Total
|
---|
g=6 | | | | | | | 1 | | | | | 1
|
---|
8 | | | | | | | 3 | | | | | 3
|
---|
11 | | | | 2 | | | | | | | 3 | 5
|
---|
41 | | 1 | 3 | | | 9 | | | | 27 | | 40
|
---|
42 | | 1 | 3 | | | 9 | | | | 27 | | 40
|
---|
52 | 2 | | | | 6 | | | 18 | 54 | | | 80
|
---|
54 | | | | | | | 26 | | | 54 | | 80
|
---|
56 | | 2 | 6 | | | 18 | 78 | | | 216 | | 320
|
---|
64 | | | | 4 | 12 | | | 36 | 108 | | 240 | 400
|
---|
Total | 2 | 4 | 12 | 6 | 18 | 36 | 108 | 54 | 162 | 324 | 243 | 969
|
---|