Summary Tables for 2-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
|
---|
8 | | | 2 | | | | 2
|
---|
12 | | 26 | | 3 | 4 | 1 | 34
|
---|
14 | | | | | | 3 | 3
|
---|
16 | | | | | 20 | 8 | 28
|
---|
18 | | 28 | | 37 | | 16 | 81
|
---|
20 | | | | | 52 | 28 | 80
|
---|
22 | | | | | 80 | 64 | 144
|
---|
24 | | | | 320 | | 136 | 456
|
---|
25 | | | | | | 128 | 128
|
---|
26 | | | | | | 128 | 128
|
---|
27 | | | | 304 | | | 304
|
---|
28 | | | | | | 256 | 256
|
---|
29 | | | | | | 256 | 256
|
---|
30 | | | | 304 | | 256 | 560
|
---|
32 | | | | | | 544 | 544
|
---|
33 | | | | 376 | | 512 | 888
|
---|
34 | | | | | | 512 | 512
|
---|
35 | | | | | | 1088 | 1088
|
---|
Total | 1 | 54 | 2 | 1344 | 156 | 3936 | 5493
|
---|
By Galois Group and Ramification Index
| e=1 | e=2 | e=3 | e=4 | e=6 | e=12 | Total
|
---|
g=1 | 1 | 3 | | 8 | | | 12
|
---|
2 | | 3 | | 4 | | | 7
|
---|
3 | | | | | 3 | | 3
|
---|
4 | | | | 1 | | | 1
|
---|
5 | | | 1 | | 3 | | 4
|
---|
6 | | | | 7 | | | 7
|
---|
7 | | 1 | | 6 | | | 7
|
---|
8 | | | | | | 3 | 3
|
---|
9 | | | | | 3 | | 3
|
---|
10 | | | | | | 4 | 4
|
---|
11 | | | | | | 8 | 8
|
---|
12 | | | | | | 4 | 4
|
---|
13 | | | | | | 4 | 4
|
---|
14 | | 4 | | 32 | | | 36
|
---|
15 | | | | | 4 | | 4
|
---|
18 | | | | | 3 | | 3
|
---|
19 | | | 1 | | 3 | | 4
|
---|
21 | | | | | 9 | | 9
|
---|
22 | | | | | | 9 | 9
|
---|
23 | | | | | | 18 | 18
|
---|
24 | | | | | | 18 | 18
|
---|
25 | | 3 | | 18 | | | 21
|
---|
26 | | | | 14 | | | 14
|
---|
27 | | | | | | 12 | 12
|
---|
28 | | | | | | 24 | 24
|
---|
29 | | 4 | | 8 | | | 12
|
---|
30 | | | | | 12 | | 12
|
---|
31 | | | | 2 | | | 2
|
---|
42 | | | | | 8 | | 8
|
---|
43 | | | | | | 1 | 1
|
---|
45 | | | | 3 | | | 3
|
---|
48 | | | | | | 36 | 36
|
---|
49 | | | | | | 24 | 24
|
---|
50 | | | | | 12 | | 12
|
---|
51 | | 4 | | 32 | | | 36
|
---|
52 | | | | | | 12 | 12
|
---|
53 | | | | | | 24 | 24
|
---|
54 | | | | | | 12 | 12
|
---|
55 | | | | 14 | | | 14
|
---|
58 | | 2 | | 12 | | | 14
|
---|
59 | | | | 14 | | | 14
|
---|
60 | | | | 14 | | | 14
|
---|
61 | | | | 14 | | | 14
|
---|
62 | | | | | | 6 | 6
|
---|
63 | | | | | | 6 | 6
|
---|
64 | | | | | | 6 | 6
|
---|
65 | | | | | | 6 | 6
|
---|
66 | | | | | | 3 | 3
|
---|
67 | | | | | | 1 | 1
|
---|
68 | | | | | | 3 | 3
|
---|
69 | | | | | 1 | | 1
|
---|
86 | | | | | | 72 | 72
|
---|
87 | | 6 | | 36 | | | 42
|
---|
88 | | | | 42 | | | 42
|
---|
89 | | | | 42 | | | 42
|
---|
92 | | | | 42 | | | 42
|
---|
94 | | | | 48 | | | 48
|
---|
95 | | | | | | 36 | 36
|
---|
96 | | | | | | 18 | 18
|
---|
97 | | | | | | 18 | 18
|
---|
98 | | | | | | 48 | 48
|
---|
99 | | | | 24 | | | 24
|
---|
100 | | | | | | 9 | 9
|
---|
101 | | | | | | 18 | 18
|
---|
102 | | | | | | 12 | 12
|
---|
103 | | | | | | 18 | 18
|
---|
104 | | | | 64 | | | 64
|
---|
105 | | 8 | | 16 | | | 24
|
---|
106 | | | | | 3 | | 3
|
---|
107 | | | | | 4 | | 4
|
---|
108 | | | | | | 36 | 36
|
---|
109 | | | | | | 36 | 36
|
---|
110 | | | | | | 36 | 36
|
---|
111 | | | | | | 36 | 36
|
---|
112 | | | | | | 36 | 36
|
---|
113 | | | | | | 36 | 36
|
---|
114 | | | | | | 36 | 36
|
---|
115 | | | | | | 36 | 36
|
---|
126 | | | | | 5 | | 5
|
---|
128 | | | | | | 5 | 5
|
---|
129 | | | | 5 | | | 5
|
---|
134 | | 16 | | 128 | | | 144
|
---|
135 | | | | | 8 | | 8
|
---|
136 | | | | | | 72 | 72
|
---|
137 | | | | | | 72 | 72
|
---|
138 | | | | | | 72 | 72
|
---|
139 | | | | | | 24 | 24
|
---|
140 | | | | | | 72 | 72
|
---|
141 | | | | 144 | | | 144
|
---|
142 | | | | 144 | | | 144
|
---|
143 | | | | 384 | | | 384
|
---|
145 | | | | | | 48 | 48
|
---|
146 | | | | | | 48 | 48
|
---|
147 | | | | | | 48 | 48
|
---|
148 | | | | | | 24 | 24
|
---|
149 | | | | | | 48 | 48
|
---|
150 | | | | | | 96 | 96
|
---|
151 | | | | | | 48 | 48
|
---|
152 | | | | | | 48 | 48
|
---|
153 | | | | | | 48 | 48
|
---|
154 | | | | | | 48 | 48
|
---|
155 | | | | | | 48 | 48
|
---|
158 | | | | | 15 | | 15
|
---|
159 | | | | | 20 | | 20
|
---|
166 | | | | 7 | | | 7
|
---|
185 | | | | | | 288 | 288
|
---|
186 | | | | | | 288 | 288
|
---|
193 | | | | | | 288 | 288
|
---|
205 | | | | 15 | | | 15
|
---|
206 | | | | | | 15 | 15
|
---|
208 | | | | | 40 | | 40
|
---|
221 | | | | | | 192 | 192
|
---|
223 | | | | | | 192 | 192
|
---|
224 | | | | | | 192 | 192
|
---|
225 | | | | | | 192 | 192
|
---|
250 | | | | | | 576 | 576
|
---|
254 | | | | | | 63 | 63
|
---|
Total | 1 | 54 | 2 | 1344 | 156 | 3936 | 5493
|
---|
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=12 | 18 | Total
|
---|
g=1 | 1 | 2 | 3
|
---|
2 | 1 | 2 | 3
|
---|
7 | 1 | | 1
|
---|
14 | 2 | 2 | 4
|
---|
25 | 1 | 2 | 3
|
---|
29 | 2 | 2 | 4
|
---|
51 | 2 | 2 | 4
|
---|
58 | 2 | | 2
|
---|
87 | 2 | 4 | 6
|
---|
105 | 4 | 4 | 8
|
---|
134 | 8 | 8 | 16
|
---|
Total | 26 | 28 | 54
|
---|
Ramification Index = 3
| d=8 | Total
|
---|
g=5 | 1 | 1
|
---|
19 | 1 | 1
|
---|
Total | 2 | 2
|
---|
Ramification Index = 4
| d=12 | 18 | 24 | 27 | 30 | 33 | Total
|
---|
g=1 | | | | | | 8 | 8
|
---|
2 | | | 4 | | | | 4
|
---|
4 | | 1 | | | | | 1
|
---|
6 | | 3 | 4 | | | | 7
|
---|
7 | | 2 | 4 | | | | 6
|
---|
14 | | 2 | 2 | 8 | 8 | 12 | 32
|
---|
25 | | 2 | 16 | | | | 18
|
---|
26 | | 2 | 12 | | | | 14
|
---|
29 | | | | | | 8 | 8
|
---|
31 | | | 2 | | | | 2
|
---|
45 | 1 | | 2 | | | | 3
|
---|
51 | | 2 | 2 | 8 | 8 | 12 | 32
|
---|
55 | | | 14 | | | | 14
|
---|
58 | | | 4 | | 8 | | 12
|
---|
59 | | 2 | 4 | 8 | | | 14
|
---|
60 | | | 6 | | 8 | | 14
|
---|
61 | | | 6 | 8 | | | 14
|
---|
87 | | | 12 | | 24 | | 36
|
---|
88 | | 2 | 16 | 24 | | | 42
|
---|
89 | | | 18 | | 24 | | 42
|
---|
92 | | | 18 | 24 | | | 42
|
---|
94 | | | 16 | | | 32 | 48
|
---|
99 | | 4 | 4 | | | 16 | 24
|
---|
104 | | | 32 | | | 32 | 64
|
---|
105 | | | | | | 16 | 16
|
---|
129 | 1 | | 4 | | | | 5
|
---|
134 | | | 16 | 32 | 32 | 48 | 128
|
---|
141 | | | 32 | 32 | 32 | 48 | 144
|
---|
142 | | 8 | 24 | 32 | 32 | 48 | 144
|
---|
143 | | | 32 | 128 | 128 | 96 | 384
|
---|
166 | | 7 | | | | | 7
|
---|
205 | 1 | | 14 | | | | 15
|
---|
Total | 3 | 37 | 320 | 304 | 304 | 376 | 1344
|
---|
Ramification Index = 6
| d=12 | 16 | 20 | 22 | Total
|
---|
g=3 | | 1 | | 2 | 3
|
---|
5 | | 1 | | 2 | 3
|
---|
9 | 1 | | 2 | | 3
|
---|
15 | | 2 | | 2 | 4
|
---|
18 | | 1 | | 2 | 3
|
---|
19 | | 1 | | 2 | 3
|
---|
21 | | 1 | 2 | 6 | 9
|
---|
30 | 1 | 1 | 4 | 6 | 12
|
---|
42 | | 4 | | 4 | 8
|
---|
50 | | 2 | 4 | 6 | 12
|
---|
69 | | | 1 | | 1
|
---|
106 | | | 1 | 2 | 3
|
---|
107 | | | 2 | 2 | 4
|
---|
126 | 1 | | 4 | | 5
|
---|
135 | | | 4 | 4 | 8
|
---|
158 | | 1 | 4 | 10 | 15
|
---|
159 | 1 | 1 | 8 | 10 | 20
|
---|
208 | | 4 | 16 | 20 | 40
|
---|
Total | 4 | 20 | 52 | 80 | 156
|
---|
Ramification Index = 12
| d=12 | 14 | 16 | 18 | 20 | 22 | 24 | 25 | 26 | 28 | 29 | 30 | 32 | 33 | 34 | 35 | Total
|
---|
g=8 | | 1 | | | | | | | 2 | | | | | | | | 3
|
---|
10 | | | | | | | | | 4 | | | | | | | | 4
|
---|
11 | | | | | | | | | | | | | | | | 8 | 8
|
---|
12 | | | | | | | | | | | | | | | | 4 | 4
|
---|
13 | | | | | 2 | | | | 2 | | | | | | | | 4
|
---|
22 | | | | 1 | | | 2 | | 2 | 4 | | | | | | | 9
|
---|
23 | | | | 2 | | | 8 | | | 8 | | | | | | | 18
|
---|
24 | | | | 2 | | | 8 | | | 8 | | | | | | | 18
|
---|
27 | | 1 | | 1 | | | 2 | | 4 | 4 | | | | | | | 12
|
---|
28 | | | | | | | | | | | 8 | | 8 | | | 8 | 24
|
---|
43 | | | | | 1 | | | | | | | | | | | | 1
|
---|
48 | | | | | | | | | 20 | 16 | | | | | | | 36
|
---|
49 | | | | | 4 | | | | 12 | 8 | | | | | | | 24
|
---|
52 | | | | | 2 | | 4 | | 6 | | | | | | | | 12
|
---|
53 | | | | | | | | | | | | | | | | 24 | 24
|
---|
54 | | | | | | | | | | | | | | | | 12 | 12
|
---|
62 | | | 1 | | | 1 | | | | | | | 4 | | | | 6
|
---|
63 | | | 1 | | | 1 | | | | | | | 4 | | | | 6
|
---|
64 | | | 1 | | | 1 | | | | | | | 4 | | | | 6
|
---|
65 | | | 1 | | | 1 | | | | | | | 4 | | | | 6
|
---|
66 | | | | | | 1 | | | 2 | | | | | | | | 3
|
---|
67 | | | | | | 1 | | | | | | | | | | | 1
|
---|
68 | | | | | | 3 | | | | | | | | | | | 3
|
---|
86 | | | | | | | | | | | 24 | | 24 | | | 24 | 72
|
---|
95 | | | | 2 | | 2 | 8 | | | | | | 24 | | | | 36
|
---|
96 | | | | 1 | | 1 | 4 | | | | | | 12 | | | | 18
|
---|
97 | | | | 1 | | 1 | 4 | | | | | | 12 | | | | 18
|
---|
98 | | | 2 | 2 | | 4 | 8 | | | | | | 32 | | | | 48
|
---|
100 | | | | | | 1 | 2 | | 2 | 4 | | | | | | | 9
|
---|
101 | | | | | | 2 | 8 | | | 8 | | | | | | | 18
|
---|
102 | | | | | | 2 | 2 | | 4 | 4 | | | | | | | 12
|
---|
103 | | | | | | 2 | 8 | | | 8 | | | | | | | 18
|
---|
108 | | | | | | 2 | 2 | | | 8 | | 8 | | | 16 | | 36
|
---|
109 | | | | | | 2 | 2 | | | 8 | | 8 | | | 16 | | 36
|
---|
110 | | | | | 2 | 2 | | 8 | | | | 8 | | 16 | | | 36
|
---|
111 | | | | | 2 | 2 | | 8 | | | | 8 | | 16 | | | 36
|
---|
112 | | | | | | 2 | 2 | | | | | 8 | 8 | | 16 | | 36
|
---|
113 | | | | | | 2 | 2 | | | | | 8 | 8 | | 16 | | 36
|
---|
114 | | | | | 2 | 2 | | 8 | | | | | 8 | 16 | | | 36
|
---|
115 | | | | | 2 | 2 | | 8 | | | | | 8 | 16 | | | 36
|
---|
128 | | 1 | | | | | | | 4 | | | | | | | | 5
|
---|
136 | | | | | | | | | 8 | 16 | | 16 | | | 32 | | 72
|
---|
137 | | | | | | | 8 | 16 | | | | 16 | | 32 | | | 72
|
---|
138 | | | | | | | | | 8 | | | 16 | 16 | | 32 | | 72
|
---|
139 | | | | | | | | | 8 | 16 | | | | | | | 24
|
---|
140 | | | | | | | 8 | 16 | | | | | 16 | 32 | | | 72
|
---|
145 | | | | | | 4 | 4 | | 8 | 32 | | | | | | | 48
|
---|
146 | | | | | 4 | 4 | 8 | | | | | 32 | | | | | 48
|
---|
147 | | | | | | 4 | 4 | | 8 | | | | 32 | | | | 48
|
---|
148 | | | | | | | 4 | | 12 | 8 | | | | | | | 24
|
---|
149 | | | | | 4 | 4 | 8 | | | | | | 32 | | | | 48
|
---|
150 | | | | | | | | | | | | | | | | 96 | 96
|
---|
151 | | | | | | | | | | | | | | | | 48 | 48
|
---|
152 | | | | | | | | | | | | | | | | 48 | 48
|
---|
153 | | | | | | | | | | | | | | | | 48 | 48
|
---|
154 | | | | | | | | | | | | | | | | 48 | 48
|
---|
155 | | | | | | | | | | | | | | | | 48 | 48
|
---|
185 | | | | | | | | | | | 32 | | 32 | 64 | 64 | 96 | 288
|
---|
186 | | | | | | | | | | | 32 | | 96 | 64 | | 96 | 288
|
---|
193 | | | | | | | | | | | 96 | | 32 | | 64 | 96 | 288
|
---|
206 | | | | | 3 | | | | 12 | | | | | | | | 15
|
---|
221 | | | | | | | | | | 32 | | 32 | 32 | | | 96 | 192
|
---|
223 | | | | | | | | | | | | 64 | | | 128 | | 192
|
---|
224 | | | | | | | | 64 | | | | | | 128 | | | 192
|
---|
225 | | | | | | | | | | 32 | | 32 | 32 | | | 96 | 192
|
---|
250 | | | | | | | | | | | 64 | | 64 | 128 | 128 | 192 | 576
|
---|
254 | 1 | | 2 | 4 | | 8 | 16 | | | 32 | | | | | | | 63
|
---|
Total | 1 | 3 | 8 | 16 | 28 | 64 | 136 | 128 | 128 | 256 | 256 | 256 | 544 | 512 | 512 | 1088 | 3936
|
---|