Summary Tables for 3-adic Fields of Degree 12
By Discriminant and Ramification Index
| e=1 | e=2 | e=3 | e=4 | e=6 | e=12 | Total
|
---|
d=0 | 1 | | | | | | 1
|
---|
6 | | 2 | | | | | 2
|
---|
9 | | | | 2 | | | 2
|
---|
12 | | | 23 | | 5 | 2 | 30
|
---|
13 | | | | | | 6 | 6
|
---|
14 | | | | | 15 | | 15
|
---|
15 | | | | | | 16 | 16
|
---|
16 | | | 46 | | | 6 | 52
|
---|
18 | | | | | 88 | 18 | 106
|
---|
19 | | | | | | 48 | 48
|
---|
20 | | | 24 | | 39 | | 63
|
---|
21 | | | | | | 114 | 114
|
---|
22 | | | | | 96 | 54 | 150
|
---|
23 | | | | | | 180 | 180
|
---|
Total | 1 | 2 | 93 | 2 | 243 | 444 | 785
|
---|
By Galois Group and Ramification Index
| e=1 | e=2 | e=3 | e=4 | e=6 | e=12 | Total
|
---|
g=1 | 1 | 1 | 3 | | 3 | | 8
|
---|
2 | | 1 | | | 3 | | 4
|
---|
3 | | | | | 6 | | 6
|
---|
5 | | | 1 | | 1 | | 2
|
---|
11 | | | 5 | | 5 | | 10
|
---|
12 | | | | | | 2 | 2
|
---|
13 | | | | | | 5 | 5
|
---|
14 | | | | 2 | | 6 | 8
|
---|
15 | | | | | | 5 | 5
|
---|
16 | | | | | 9 | | 9
|
---|
17 | | | 2 | | 2 | | 4
|
---|
18 | | | | | 24 | | 24
|
---|
19 | | | 4 | | 4 | | 8
|
---|
34 | | | | | 8 | | 8
|
---|
35 | | | | | | 8 | 8
|
---|
36 | | | | | | 8 | 8
|
---|
38 | | | | | | 10 | 10
|
---|
39 | | | 4 | | 4 | | 8
|
---|
40 | | | | | 4 | | 4
|
---|
41 | | | 2 | | 2 | | 4
|
---|
42 | | | | | | 40 | 40
|
---|
46 | | | 4 | | | | 4
|
---|
47 | | | | | 4 | | 4
|
---|
70 | | | | | 36 | | 36
|
---|
71 | | | | | 4 | | 4
|
---|
72 | | | 2 | | 2 | | 4
|
---|
73 | | | 8 | | 8 | | 16
|
---|
84 | | | | | | 16 | 16
|
---|
116 | | | | | | 20 | 20
|
---|
118 | | | | | | 8 | 8
|
---|
119 | | | 10 | | 10 | | 20
|
---|
120 | | | | | | 20 | 20
|
---|
121 | | | | | | 32 | 32
|
---|
130 | | | | | 32 | | 32
|
---|
131 | | | 16 | | 16 | | 32
|
---|
167 | | | | | | 160 | 160
|
---|
169 | | | | | | 40 | 40
|
---|
170 | | | 16 | | 16 | | 32
|
---|
171 | | | | | 8 | | 8
|
---|
172 | | | | | 6 | | 6
|
---|
173 | | | 16 | | | | 16
|
---|
174 | | | | | 6 | | 6
|
---|
212 | | | | | | 64 | 64
|
---|
215 | | | | | 20 | | 20
|
---|
Total | 1 | 2 | 93 | 2 | 243 | 444 | 785
|
---|
By Galois Group and Discriminant
Ramification Index = 1
| d=0 | Total
|
---|
g=1 | 1 | 1
|
---|
Total | 1 | 1
|
---|
By Galois Group and Discriminant
Ramification Index = 2
| d=6 | Total
|
---|
g=1 | 1 | 1
|
---|
2 | 1 | 1
|
---|
Total | 2 | 2
|
---|
Ramification Index = 3
| d=12 | 16 | 20 | Total
|
---|
g=1 | | 3 | | 3
|
---|
5 | | 1 | | 1
|
---|
11 | 2 | | 3 | 5
|
---|
17 | | 2 | | 2
|
---|
19 | | 4 | | 4
|
---|
39 | 1 | | 3 | 4
|
---|
41 | 2 | | | 2
|
---|
46 | 2 | 2 | | 4
|
---|
72 | | 2 | | 2
|
---|
73 | | 8 | | 8
|
---|
119 | 4 | | 6 | 10
|
---|
131 | | 16 | | 16
|
---|
170 | 4 | | 12 | 16
|
---|
173 | 8 | 8 | | 16
|
---|
Total | 23 | 46 | 24 | 93
|
---|
Ramification Index = 4
| d=9 | Total
|
---|
g=14 | 1 | 1
|
---|
Total | 1 | 1
|
---|
Ramification Index = 6
| d=12 | 14 | 18 | 20 | 22 | Total
|
---|
g=1 | | | 3 | | | 3
|
---|
2 | | | 3 | | | 3
|
---|
3 | | 2 | 1 | | 3 | 6
|
---|
5 | | | 1 | | | 1
|
---|
11 | | 2 | | | 3 | 5
|
---|
16 | | 1 | 2 | | 6 | 9
|
---|
17 | | 2 | | | | 2
|
---|
18 | | 2 | 10 | | 12 | 24
|
---|
19 | | | 4 | | | 4
|
---|
34 | 2 | | | 6 | | 8
|
---|
39 | | 1 | | | 3 | 4
|
---|
40 | | 2 | 2 | | | 4
|
---|
41 | | | 2 | | | 2
|
---|
47 | 1 | | | 3 | | 4
|
---|
70 | | 1 | 11 | | 24 | 36
|
---|
71 | | | 1 | | 3 | 4
|
---|
72 | | | 2 | | | 2
|
---|
73 | | 2 | 6 | | | 8
|
---|
119 | | | 4 | | 6 | 10
|
---|
130 | | | 8 | | 24 | 32
|
---|
131 | | | 16 | | | 16
|
---|
170 | | | 4 | | 12 | 16
|
---|
171 | | | 8 | | | 8
|
---|
172 | | | | 6 | | 6
|
---|
174 | | | | 6 | | 6
|
---|
215 | 2 | | | 18 | | 20
|
---|
Total | 5 | 15 | 88 | 39 | 96 | 243
|
---|
Ramification Index = 12
| d=12 | 13 | 15 | 16 | 18 | 19 | 21 | 22 | 23 | Total
|
---|
g=12 | | | | | | 2 | | | | 2
|
---|
13 | | | 2 | | | | | | 3 | 5
|
---|
14 | | | | | | 6 | | | | 6
|
---|
15 | | | 2 | | | | | | 3 | 5
|
---|
35 | | 2 | | | | | 6 | | | 8
|
---|
36 | | 2 | | | | | 6 | | | 8
|
---|
38 | | | | | | 4 | | | 6 | 10
|
---|
42 | | | 4 | | | 12 | | | 24 | 40
|
---|
84 | 2 | | | 2 | 6 | | | 6 | | 16
|
---|
116 | | | 2 | | | | 6 | | 12 | 20
|
---|
118 | | | | | | 2 | 6 | | | 8
|
---|
120 | | | 2 | | | | 6 | | 12 | 20
|
---|
121 | | 2 | | | | 6 | 24 | | | 32
|
---|
167 | | | 4 | | | 12 | 48 | | 96 | 160
|
---|
169 | | | | | | 4 | 12 | | 24 | 40
|
---|
212 | | | | 4 | 12 | | | 48 | | 64
|
---|
Total | 2 | 6 | 16 | 6 | 18 | 48 | 114 | 54 | 180 | 444
|
---|