Old Sextic Fields Unramified Outside of {∞, 3, 19}

Discriminant class: −19

∞  3 19 a_{1}  a_{2} a_{3}  Over p = 3  Over p = 19  2 5 7 11 13 17 23 29
G  h  a_{4}  a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
1.  3_{i}  0_{+} 5_{k}  -1  2  8 6 0_{+} ∗  1^{6} 5_{k} 19∗  6 33  6 33 33 6
C_{6}  -1  -5  7  222 222 6 33 33 6 6 33
2.  3_{i}  8_{+} 3_{k}  0  -9  2 2^{3} 8_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 222  33 33 222 33 33 222
S_{3}  63  48 20  1^{2} 1_{i} 19∗ 19∗ 222 222 222 33 33 222 222 33
3.  3_{i}  8_{+} 3_{k}  -3  12 -17 2^{3} 8_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 6 33 33 33 6  33 6
C_{6}  4  87 -114 323  1^{2} 1_{i} 19∗ 19∗ 6 222 6 33 33 222 6 33
4.  3_{i}  8_{+} 5_{k}  0  0 -76 2^{3} 8_{+} ∗  1^{6} 5_{k} 19∗  222 33 33 33 222 33  6
C_{6}  3  684 -2508 3743  6 222 6  33 6 6 33
5.  3_{i}  8_{+} 5_{k}  0  0 -95 2^{3} 8_{+} ∗  1^{6} 5_{k} 19∗  6  33 33 6 33 33 222
C_{6}  21  684 -570 2375  6 222 222 33  6 222
6.  3_{i}  2_{+} 5_{k}  0  0  0 2^{2} 2_{+} 1 V 1^{6} 5_{k} 19∗  222 22 33 22 6 22 22 222
D_{6}  3  0  0 19 2 0_{+} ∗ ∗  6 6 222 33 22 222 222 33
7.  3_{i}  10_{+} 3_{k}  3  18 25 2^{3} 10_{+} ∗  3^{2} 3_{k} 19∗  222 22 33 22 6 22 22 222
D_{6}  81 156 176  6 6 222 33 22 222 222
8.  3_{i}  10_{+} 5_{k}  0  0  0 2^{3} 10_{+} ∗  1^{6} 5_{k} 19∗  222 22 33 22 6 22 22 222
D_{6}  6  0  0 171  222 222 222  22 222 222 33
9.  3_{i}  10_{+} 5_{k}  0  0  0 2^{3} 10_{+} ∗  1^{6} 5_{k} 19∗  222 22  22 222 22 22 222
D_{6}  12  0  0 1539  6 6 222 33 22 222 222 33
10.  3_{i}  8_{+} 3_{k}  0  12  -5 2^{3} 8_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 6 33 3 3 6 33 3 6
S_{3} × C_{3}  36 -30 49  1^{2} 1_{i} 19∗ 19∗ 6 222 6 3 3 222 6 3
11.  3_{i}  8_{+} 5_{k}  0  0 38 2^{3} 8_{+} ∗  1^{6} 5_{k} 19∗  222 33 3 3 222 3 33 6
S_{3} × C_{3}  3  171 -798 1292  6 222 6 33 3 6 6 3
12.  3_{i}  8_{+} 5_{k}  0  0 38 2^{3} 8_{+} ∗  1^{6} 5_{k} 19∗  6 33 33 33 6 3 3 6
S_{3} × C_{3}  3  171 912 1577  222 222 6 3 3 6 6 3
13.  3_{i}  8_{+} 5_{k}  0  0 76 2^{3} 8_{+} ∗  1^{6} 5_{k} 19∗  6  3 3 6 3 3 222
S_{3} × C_{3}  3  171 114 1463  6 222 222 3 33 6 222 33
14.  1_{k}  8_{+} 1_{i}  0  -3  2 2^{3} 8_{+} ∗  2 0_{+} ∗  ∗ 6 33 33 33 6 22 33 6
A_{4} × C_{2}  3  -3  -1  1^{2} 1_{i} 19∗ 19∗ 6 2 6 33 33 2 6 33
15.  3_{i}  10_{+} 5_{k}  3  18 202 2^{3} 10_{+} ∗  1^{6} 5_{k} 19∗  4 22 33 42 6 42 42 4
S_{4} × C_{2}  6  1116 2127 1493  2 2 222  22 4 4 33
16.  3_{i}  10_{+} 5_{k}  -3  18 -31 2^{3} 10_{+} ∗  1^{6} 5_{k} 19∗  4 42 22 22 2 22 42 4
S_{4} × C_{2} 24  90 -75 68  6 6 4 33 22 222 222 33

Discriminant class: −3

∞  3 19 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 19  2 5 7 11 13 17 23 29
G  h  a_{4} a_{5}  a_{6} f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
17.  3_{i}  3_{k} 4_{+} -1  7  -8 3^{2} 3_{k} 3∗  1^{3} 2_{+} 1  6 6  222 33 6 6 6
C_{6}  3  43 -42  49  1^{3} 2_{+} 1  6 33 6 6 6 33
18.  3_{i}  3_{k} 4_{+} 0  1  0 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1  222 222 33 222 33 222 222 222
S_{3}  3  13  0  12 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1  33 33 222 33 222 222 222 33
19.  3_{i}  9_{k} 4_{+} 0  0  -19 1^{6} 9_{k} 3∗  1^{3} 2_{+} 1  6 222 33 6 33 6 6 222
C_{6}  9  0  0 6859  1^{3} 2_{+} 1  33  222 33 222 6 222

Discriminant class: −3 (continued)

∞  3 19 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 19  2 5 7 11 13 17 23 29
G  h  a_{4} a_{5}  a_{6} f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
20.  3_{i}  9_{k} 4_{+} 0  0 -152 1^{6} 9_{k} 3∗  1^{3} 2_{+} 1  222 6 33 6  6 222 6
C_{6}  9  0  0 6859  1^{3} 2_{+} 1  33  6  6 6 6 33
21.  3_{i}  11_{k} 4_{+} 0  0  0 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  222 222 33 222 33 222 222 222
S_{3}  36  0  0 1083  1^{3} 2_{+} 1  222  222 222 222 33
22.  3_{i}  11_{k} 4_{+} 3  6 121 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  222 222  222  222 222 222
S_{3}  36  177 -168 3136  1^{3} 2_{+} 1  33 33 222 33 222 222 222 33
23.  3_{i}  9_{k} 0_{+} 0  0  -1 1^{6} 9_{k} 3∗  6 6 33 6 33 222 6 6
C_{6}  0  0  1  33  6 33 6 222 6 33
24.  3_{i}  11_{k} 0_{+} 0  0  0 1^{6} 11_{k} 3∗  3 0_{+} 1  222 222 33 222 33 222 222 222
S_{3}  0  0  3  3 0_{+} 1  33 33 222 33 222 222 222
25.  3_{i}  9_{i} 2_{−} 0 -6  -10 1^{6} 9_{i} 3∗  1^{2} 1_{i} 19∗ 19∗ 222 222 33 6 22 6 6 222
D_{6}  36 30  25  1^{2} 1_{i} 19∗ 19∗ 22 22 222 33 6 222 222 33
26.  3_{i}  3_{k} 2_{+} -1  1  -2 3^{2} 3_{k} 3∗  1^{3} 2_{+} 1  6 6 33 222 3 6 6 6
S_{3} × C_{3}  4 -3  1  33 33 6 3 6 6 6 3
27.  3_{i}  9_{k} 2_{+} 0  0  -8 1^{6} 9_{k} 3∗  1^{3} 2_{+} 1  222 6 3 6 33 6 222 6
S_{3} × C_{3}  3  0  0  19  3 33 6 33 6 6 6 3
28.  3_{i}  9_{k} 2_{+} 0  0  -1 1^{6} 9_{k} 3∗  1^{3} 2_{+} 1  6 222 3 6 3 6 6 222
S_{3} × C_{3}  3  0  0  19  3 33 222 3 222 6 222 33
29.  3_{i}  11_{k} 2_{+} 0  0  -3 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  6 222 3 6 3 6 6 222
S_{3} × C_{3}  3  0  0 171  3 0_{+} 1  33  222 33 222 6 222 33
30.  3_{i}  11_{k} 2_{+} 0  0  -6 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  222 6 33 6  6 222 6
S_{3} × C_{3}  3  0  0  57  3 0_{+} 1  3 33 6 33 6 6 6 3
31.  3_{i}  11_{k} 2_{+} 0  0  21 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  6 6 33 222 3 6 6 6
S_{3} × C_{3}  3  0  0 171  3 0_{+} 1  6 33 6 6 6 3
32.  3_{i}  11_{k} 2_{+} 0  0  9 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  6 6  222 33 6 6 6
S_{3} × C_{3}  3  0  0  57  3 0_{+} 1  33 33 6 3 6 6 6 3
33.  3_{i}  11_{k} 2_{+} 0  0  -15 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  6 222 33 6 33 6 6 222
S_{3} × C_{3}  3  0  0  57  3 0_{+} 1  3 33 222 3 222 6 222 33
34.  3_{i}  11_{k} 2_{+} 0  0  24 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  222 6 3 6 33 6 222 6
S_{3} × C_{3}  3  0  0 171  3 0_{+} 1  33  6  6 6 6 3
35.  3_{i}  9_{k} 4_{+} 0  0  19 1^{6} 9_{k} 3∗  1^{3} 2_{+} 1  6 6 3 6 3 222 6 6
S_{3} × C_{3} 27  0  0 361  1^{3} 2_{+} 1  3 33 6 3 6 222 6 3
36.  3_{i}  11_{k} 4_{+} 0  0  57 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  6 6 3 6 3 222 6 6
S_{3} × C_{3}  9  0  0 1083  1^{3} 2_{+} 1  33  6 33 6 222 6 3
37.  3_{i}  11_{k} 4_{+} 0  0  57 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  6 6 33 6 33 222 6 6
S_{3} × C_{3}  9  0  0 3249  1^{3} 2_{+} 1  3 33 6 3 6 222 6 3
38.  3_{i}  11_{k} 4_{+} 0  0 114 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  222 6 3 6 33 6 222 6
S_{3} × C_{3} 63  0  0 20577  1^{3} 2_{+} 1  3 33 6 33 6 6 6 33
39.  3_{i}  11_{k} 4_{+} 0  0 171 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  6 6 33 222 3 6 6 6
S_{3} × C_{3}  9  0  0 20577  1^{3} 2_{+} 1  33 33 6 3 6 6 6 33
40.  3_{i}  11_{k} 4_{+} 0  0 -285 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  6 222 3 6 3 6 6 222
S_{3} × C_{3}  9  0  0 20577  1^{3} 2_{+} 1  3 33 222 3 222 6 222
41.  3_{i}  11_{k} 0_{+} 0  0  3 1^{6} 11_{k} 3∗  3 0_{+} 1  6 6 3 6 3 222 6 6
S_{3} × C_{3}  0  0  3  3 0_{+} 1  3 33 6 3 6 222 6 33
42.  1_{k}  9_{k} 2_{−} 3 -6  -17 1^{6} 9_{k} 3∗  1^{2} 1_{k} 19 19 6 6 33 6 33 2 6 6
A_{4} × C_{2}  -3  6  19  1^{2} 1_{k} 19 19 33 22 6 33 6 2 6 33

Discriminant class: −3 (continued)

∞  3 19 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 19  2 5 7 11 13 17 23 29
G  h  a_{4} a_{5}  a_{6} f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
43.  3_{i}  11_{k} 4_{+} 3  6  7 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  4 4 22 222 22 222 4 4
S_{4}^{−}  24  6  3  58  1^{3} 2_{+} 1  33 33 4 33 222 222 222 33
44.  3_{i}  11_{k} 4_{+} 3 -3  -11 1^{6} 11_{k} 3∗  1^{3} 2_{+} 1  4 222 33 4 33 4 4 4
S_{4}^{−}  24  177 696 1588  1^{3} 2_{+} 1  22 22 222  222 4 4 33
45.  3_{i}  11_{k} 4_{+} -3 15  29 1^{6} 11_{k} 3∗  1^{5} 4_{+} 1  4 4 5 6 5 4 4 6
S_{5}  8  -3 258 760  33 33 4 5 6 222 222 5

Discriminant class: 1

∞  3 19 a_{1} a_{2} a_{3}  Over p = 3  Over p = 19 2 5 7 11 13 17 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
46. 2_{−} 8_{+} 2_{−} 0 -3 10 1^{3} 4_{+} 1  1^{2} 1_{k} 19 19 33 33 33 33 33 22 33 33
A_{4}  2  -18 18 -9 1^{3} 4_{+} 1  1^{2} 1_{k} 19 19 33 22 33 33 33 22 33 33
47. 2_{−} 10_{−} 4_{+} 0 -12  0 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  42 42 22 22 22 22 42 42
S_{4}^{+}  3  48 0 -121 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  33 33 42 33 22 22 22 33
48. 2_{−} 10_{−} 4_{+} 3 15 25 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  42 22 33 42 33 42 42 42
S_{4}^{+}  3  60 48 -107 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  22 22 22  22 42 42 33
49. 2_{−} 10_{−} 2_{+} -3 3  2 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 5 33 5 5 33 33 33 5
A_{5}  -9 9 -6 1^{3} 5_{i} 3∗  1^{2} 1_{k} 19 19 5 5 5 22 22 33 22 22
50. 2_{−} 6_{−} 4_{+} -3 3  1 1^{3} 3_{i} 3∗  1^{5} 4_{+} 1  33 5 5 33 22 5 33 33
A_{5}  2  -9 3 -3 1^{3} 3_{i} 3∗  22 33 33 5 5 5 5 33
51. 2_{−} 6_{−} 4_{+} 3 6 -17 1^{3} 3_{k} 3  1^{5} 4_{+} 1  5 5 22 5 33 33 5 5
A_{5}  2  -27 -3 -21 1^{3} 3_{k} 3  33 33 22 33 22 33 33 5

Discriminant class: 57

∞  3 19 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 19  2 5 7 11 13 17 23 29
G  h  a_{4} a_{5}  a_{6} f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
52.  0_{+}  3_{i} 5_{k}  -1 -17 -11 3^{2} 3_{i} 3  1^{6} 5_{k} 19∗  33 6  222 6 6 6 33
C_{6}  37 33  7  222 222 33 33 6 33 33 33
53.  0_{+}  9_{i} 3_{k}  0 -30 -14 1^{6} 9_{i} 3  1^{2} 1_{i} 19∗ 19∗ 33 6 33 6 6 222 6 33
C_{6}  225 210 -179  1^{2} 1_{i} 19∗ 19∗ 6 222 33 33 6  33 33
54.  0_{+}  9_{i} 5_{k}  0 -57 -95 1^{6} 9_{i} 3  1^{6} 5_{k} 19∗  33 222 33 6 6 6 6
C_{6}  3  684 1596 -152  6 222  33 222 33
55.  0_{+}  9_{i} 5_{k}  0 -57 -76 1^{6} 9_{i} 3  1^{6} 5_{k} 19∗  6 33 6 222 6 222 33
C_{6}  3  684 1824 1216  6 222 33  6 33 33 33
56.  2_{−}  3_{k} 5_{k}  3  -1  -7 2^{2} 2_{+} 1 V 1^{6} 5_{k} 19∗  22 222 33 222 6 222 222 22
D_{6}  3  -5  -1  -8 1^{2} 1_{k} 3 3  6 6 22 33 222 22 22 33
57.  2_{−}  9_{k} 3_{k}  0  0  5 1^{6} 9_{k} 3  1^{2} 1_{i} 19∗ 19∗ 22 222 33 6 222 6 6 22
D_{6}  0  0  -8  1^{2} 1_{i} 19∗ 19∗ 222 222 22 33 6 22 22 33
58.  2_{−}  11_{k} 3_{k}  0  0 -45 1^{6} 11_{k} 3  3^{2} 3_{k} 19∗  22 222 33 222 6 222 222 22
D_{6}  0  0 -192  6 6 22 33 222 22 22
59.  2_{−}  11_{k} 5_{k}  0  0  0 1^{6} 11_{k} 3  1^{6} 5_{k} 19∗  22 222 33 222 6 222 222 22
D_{6}  6  0  0 -4617  222 222 22  222 22 22 33
60.  2_{−}  11_{k} 5_{k}  0  0  0 1^{6} 11_{k} 3  1^{6} 5_{k} 19∗  22 222  222 222 222 222 22
D_{6}  12  0  0 -57  6 6 22 33 222 22 22 33

Discriminant class: 57 (continued)

∞  3 19 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 19  2 5 7 11 13 17 23 29
G  h  a_{4} a_{5}  a_{6} f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
61.  2_{−}  9_{i} 1_{i}  3  3  5 1^{6} 9_{i} 3  2 0_{+} ∗  ∗ 33 6 33 6 6 2 6 33
A_{4} × C_{2}  3  3  1  1^{2} 1_{i} 19∗ 19∗ 6 2 33 33 6 22 33 33
62.  2_{−}  11_{k} 3_{k}  0  0 -15 1^{6} 11_{k} 3  3^{2} 3_{k} 19∗  22 222 3 6 6 6 6 22
S_{3} × S_{3}  0  0 -72  6 6 22 3 6 22 22 33
63.  2_{−}  9_{k} 5_{k}  0  0 -76 1^{6} 9_{k} 3  1^{6} 5_{k} 19∗  22 222 3 6 6 6 6 22
S_{3} × S_{3}  9  0  0  19  6 6 22 3 6 22 22 3
64.  2_{−}  11_{k} 5_{k}  0  0 -228 1^{6} 11_{k} 3  1^{6} 5_{k} 19∗  22 222 3 6 6 6 6 22
S_{3} × S_{3}  3  0  0 171  222 222 22 33 6 22 22 3
65.  2_{−}  11_{k} 5_{k}  0  0 -57 1^{6} 11_{k} 3  1^{6} 5_{k} 19∗  22 222 33 6 222 6 6 22
S_{3} × S_{3}  3  0  0 456  6 6 22 3 6 22 22 3
66.  2_{−}  11_{k} 5_{k}  0  0 -57 1^{6} 11_{k} 3  1^{6} 5_{k} 19∗  42 4 22 222 2 222 4 42
S_{4} × C_{2} 24  -171 -171 -57  6 6 42 33 222 22 22 33
67.  2_{−}  11_{k} 5_{k}  3 -39 -83 1^{6} 11_{k} 3  1^{6} 5_{k} 19∗  42 222 33 4 6 4 4 42
S_{4} × C_{2}  6  204 246 -407  2 2 22  222 42 42 33
68.  2_{−}  7_{k} 3_{k}  -3  -3  18 1^{6} 7_{k} 3  1^{4} 3_{k} 19 19∗ 5 4 33 6 4 6 4 33
S_{5}  -33 27 -12  2 0_{+} ∗  ∗ 4 6 33 5 4 33 5 22
69.  2_{−}  11_{k} 3_{k}  0  9 -21 1^{6} 11_{k} 3  1^{4} 3_{k} 19 19∗ 5 222 5 6 4 4 4 22
S_{5}  27 27 -18  2 0_{+} ∗  ∗ 4 4 33 5 222 33 5 5
70.  2_{−}  11_{k} 5_{k}  -3 -39 197 1^{6} 11_{k} 3  1^{6} 5_{k} 19∗  5 6 33 4 6 6 6 5
S_{5}  12  -138 609 -293  4 6 33 5 6 5 33 5