Old Sextic Fields Unramified Outside of {∞, 3, 5}

Discriminant class: −15

∞  3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5  2 7 11 13 17 19 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}  7_{i} 3_{−} 3  3 1 1^{6} 7_{i} 3  1^{2} 1_{−} 5∗ 5∗ 33 222 222 222 33 33 33 222
S_{3}  2  3  3 1  1^{2} 1_{−} 5∗ 5∗ 33 222 222 222  33 222 33
2.  3_{i}  9_{i} 3_{−} 0  6 -4 1^{6} 9_{i} 3  3^{2} 3_{−} 5∗  33 6 6 6  33 6
C_{6}  2  9 -12 19  33 222 6 6 33  6 33
3.  3_{i}  7_{k} 5_{+} -3  0 -5 1^{6} 7_{k} 3  1^{6} 5_{+} 5∗  22 6 222 222 22 33 22 222
D_{6}  2  15 12 4  33 6 222 6 22 22 222 33
4.  3_{i}  11_{k} 3_{+} 0  0 -9 1^{6} 11_{k} 3  2^{2} 2_{−} 1 V 22 6 222 6 22 33 22 222
D_{6}  2  0  0 24  1^{2} 1_{−} 5∗ 5∗ 33 6 222 6 22 22 222
5.  3_{i}  11_{k} 5_{+} 0  0 0 1^{6} 11_{k} 3  1^{6} 5_{+} 5∗  22 222 222 6 22 33 22 222
D_{6}  4  0  0 15  6 222 6 22 22 222 33
6.  3_{i}  11_{k} 5_{+} 0  0 -15 1^{6} 11_{k} 3  1^{6} 5_{+} 5∗  22 6 222 6 22  22 222
D_{6}  2  0  0 360  33 222 222 222 22 22 222 33
7.  3_{i}  9_{i} 3_{−} 0  6 1 1^{6} 9_{i} 3  3^{2} 3_{−} 5∗  3 6 6 6 33 33 3 6
S_{3} × C_{3}  2  9  3 4  3 222 6 6 33 33 6 3
8.  3_{i}  11_{k} 5_{+} 0  0 -15 1^{6} 11_{k} 3  1^{6} 5_{+} 5∗  42 2 4 6 42 33 22 4
S_{4} × C_{2}  4  45 -45 15  22 6 4 6 42 42 4 33
9.  3_{i}  9_{k} 5_{+} 0  0 -5 1^{6} 9_{k} 3  1^{5} 5_{+} 5∗  5 4 4 4 33 33 5 4
S_{5}  2  0  9 10  5 4 6 6 5 22 6 5
10.  3_{i}  7_{k} 7_{+} 3 15 15 1^{6} 7_{k} 3  1^{5} 7_{+} 5∗  33 4 6 6 5 5 22 4
S_{5}  2  45 -9 63  5 6 6 4 5 33 6 5
11.  3_{i}  11_{k} 5_{+} 0  0 -15 1^{6} 11_{k} 3  1^{5} 5_{+} 5∗  5 6 4 222 22 5 33 6
S_{5}  2  0  9 60  22 6 4 4 33 33 6 5
12.  3_{i}  9_{k} 7_{+} 0  0 0 1^{6} 9_{k} 3  1^{5} 7_{+} 5∗  33 6 6 6 33 22 5 4
S_{5}  2  135 -162 135  5 4 4 222 22 5 4 33

Discriminant class: −3

∞  3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 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
13.  3_{i}  7_{k} 4_{+} 3 6 17 1^{6} 7_{k} 3∗  2^{3} 4_{+} ∗  222 33 222  222 33 222 222
S_{3}  21 -12 16  33 33 222 33 222 222 222 33
14.  3_{i}  11_{k} 4_{+} 3 6 37 1^{6} 11_{k} 3∗  2^{3} 4_{+} ∗  222  222 33 222 33 222 222
S_{3}  4  51 -42 196  33 222 33 222 222 222 33
15.  3_{i}  11_{k} 4_{+} 0 0 0 1^{6} 11_{k} 3∗  2^{3} 4_{+} ∗  222 33 222 33 222  222 222
S_{3}  0 0 75  33  222  222 222 222 33
16.  3_{i}  9_{k} 0_{+} 0 0 -1 1^{6} 9_{k} 3∗  6 0_{+} ∗  6 33 6 33 222  6 6
C_{6}  0 0 1  33  6 33 6 222 6 33
17.  3_{i}  11_{k} 0_{+} 0 0 0 1^{6} 11_{k} 3∗  2 0_{+} ∗ ∗ 222 33 222 33 222 33 222 222
S_{3}  0 0 3  2 0_{+} ∗ ∗ 33 33 222 33 222 222 222
18.  3_{i}  7_{i} 2_{−} 3 3 0 1^{6} 7_{i} 3∗  2^{2} 2_{−} 1 V 6 22 222 22 6 33 6 222
D_{6}  -3 0 3  2 0_{+} ∗ ∗ 33 22 222 22 222 6 222 33
19.  3_{i}  9_{k} 4_{+} 0 0 5 1^{6} 9_{k} 3∗  2^{3} 4_{+} ∗  6 3 6 33 222 33 6 6
S_{3} × C_{3}  0 0 25  3 33 6 3 6 222 6 3

Discriminant class: −3 (continued)

∞  3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 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}  11_{k} 4_{+} 0 0 15 1^{6} 11_{k} 3∗  2^{3} 4_{+} ∗  6 3 6 3 222  6 6
S_{3} × C_{3}  0 0 75  3  6 33 6 222 6 3
21.  3_{i}  11_{k} 4_{+} 0 0 15 1^{6} 11_{k} 3∗  2^{3} 4_{+} ∗  6 33 6 3 222 33 6 6
S_{3} × C_{3}  0 0 225  33 33 6 3 6 222 6 3
22.  3_{i}  11_{k} 0_{+} 0 0 3 1^{6} 11_{k} 3∗  6 0_{+} ∗  6 3 6 3 222 33 6 6
S_{3} × C_{3}  0 0 3  3 33 6 3 6 222 6 33
23.  3_{i}  11_{k} 4_{+} 3 6 22 1^{6} 11_{k} 3∗  2^{3} 4_{+} ∗  4 22 4 33 4 33 222 4
S_{4}^{−}  8  51 48 16  22 33 4 33 4 4 4 33
24.  3_{i}  3_{k} 8_{+} 0 -5 0 1^{4} 3_{k} 3 3∗ 1^{5} 8_{+} ∗  6 5 6 5 6 22 4 4
S_{5}  15 -11 5 2 0_{+} ∗ ∗  33 33 4 33 4 4 6 5
25.  3_{i}  9_{k} 6_{+} 0 0 -10 1^{6} 9_{k} 3∗  1^{5} 6_{+} ∗  4 33 6 5 4 33 4 6
S_{5}  12  0 9 10  5 5 4 5 6 222 6 33
26.  3_{i}  11_{k} 6_{+} 3 15 -5 1^{6} 11_{k} 3∗  1^{5} 6_{+} ∗  4 5 4 5 6 22 4 6
S_{5}  4  15 39 22  22 22 6 33 6 4 6 33
27.  3_{i}  11_{k} 8_{+} 3 15 100 1^{6} 11_{k} 3∗  1^{5} 8_{+} ∗  6 22 6 5 4 22 6 4
S_{5}  8  375 660 505  5 22 222 33 4 4 6 33

Discriminant class: 1

∞  3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 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
28. 2_{−} 10_{−} 4_{+} 3  3  1 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  42 22 42 33 42 33 22 42
S_{4}^{+}  0  0 -15 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  22 33 42 33 42 42 42 33
29. 2_{−} 6_{−} 8_{+} 0  0 -25 1^{3} 3_{k} 3  1^{5} 8_{+} 1  5 33 5 5 33 33 22 5
A_{5}  2  0 -15 150 1^{3} 3_{k} 3  33 22 5 5 33 5 5 33
30. 2_{−} 6_{−} 8_{+} 0 -15 -35 1^{3} 3_{i} 3∗  1^{5} 8_{+} 1  5 5 33 22 33 33 33 5
A_{5}  2  -45 -51 -30 1^{3} 3_{i} 3∗  22 5 33 22 5 5 22 33
31. 2_{−} 10_{−} 6_{+} -3 -15 50 1^{3} 5_{i} 3∗  1^{5} 6_{+} 1  5 5 5 5 5 33 5 22
A_{5}  2  -45 27 -36 1^{3} 5_{i} 3∗  33 5 5 22 22 33 33 5
32. 2_{−} 10_{−} 6_{+} 3 15 55 1^{3} 5_{i} 3∗  1^{5} 6_{+} 1  33 5 5 22 33 33 22 5
A_{5}  2  105 273 169 1^{3} 5_{i} 3∗  5 33 22 5 5 5 33 5
33. 2_{−} 10_{−} 8_{+} 0 15 -45 1^{3} 5_{i} 3∗  1^{5} 8_{+} 1  5 33 22 33 5 22 5 5
A_{5}  2  30 -18  5 1^{3} 5_{i} 3∗  33 33 33 5 5  5 33
34. 2_{−} 10_{−} 8_{+} 0 -30 -75 1^{3} 5_{i} 3∗  1^{5} 8_{+} 1  33 22 5 33 5 5 5 22
A_{5}  2  -60 -549 -895 1^{3} 5_{i} 3∗  33 5 33 33 33 22 22 22

Discriminant class: 5

∞  3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 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
35.  0_{+}  8_{+} 3_{+}  0 -9  4 2^{3} 8_{+} ∗  3^{2} 3_{+} 5  6 6 33 6 222  6 33
C_{6}  9 -3 -1  33 222 33 6 6 222 33 33
36.  2_{−}  6_{+} 3_{−}  0  0  1 2^{3} 6_{+} ∗  2^{2} 2_{−} 1 V 6 222 22 222 6 33 6 22
D_{6}  0  0 -1  1^{2} 1_{+} 5 5 33 222 22 222 222 6 22 33
37.  2_{−}  6_{+} 5_{−}  0  0  0 2^{3} 6_{+} ∗  1^{6} 5_{−} 5  222 6 22 222 222 33 222 22
D_{6}  0  0 -5  33 6 22 6 222 222 22 33

Discriminant class: 5 (continued)

∞  3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 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
38.  2_{−}  10_{+} 3_{−}  0  0 12 2^{3} 10_{+} ∗  2^{2} 2_{−} 1 V 222 6 22 6 222 33 222 22
D_{6}  0  0 -9  1^{2} 1_{+} 5 5 33 6 22 6 222 222 22
39.  2_{−}  10_{+} 5_{−}  0  0  0 2^{3} 10_{+} ∗  1^{6} 5_{−} 5  222 6 22 6 222  222 22
D_{6}  0  0 -45  33 222 22 222 222 222 22 33
40.  2_{−}  10_{+} 5_{−}  0  0  0 2^{3} 10_{+} ∗  1^{6} 5_{−} 5  222 222 22 6 222 33 222 22
D_{6}  2  0  0 -405  6 22 6 222 222 22 33
41.  2_{−}  6_{+} 5_{−}  0  0 -5 2^{3} 6_{+} ∗  1^{6} 5_{−} 5  6 6 22 222 6 3 6 22
S_{3} × S_{3}  0  0  5  3 6 22 6 222 6 22 3
42.  2_{−}  10_{+} 3_{−}  0  0 -3 2^{3} 10_{+} ∗  2^{2} 2_{−} 1 V 6 6 22 6 6 3 6 22
S_{3} × S_{3}  0  0 -9  1^{2} 1_{+} 5 5 3 6 22 6 222 6 22 33
43.  2_{−}  10_{+} 5_{−}  0  0 -15 2^{3} 10_{+} ∗  1^{6} 5_{−} 5  6 6 22 6 6 33 6 22
S_{3} × S_{3}  0  0 45  3 222 22 222 222 6 22 3
44.  2_{−}  10_{+} 5_{−}  0  0 15 2^{3} 10_{+} ∗  1^{6} 5_{−} 5  6 222 22 6 6 3 6 22
S_{3} × S_{3}  0  0 -225  33 6 22 6 222 6 22 3
45.  2_{−}  10_{+} 5_{−} -3  0 -10 2^{3} 10_{+} ∗  1^{6} 5_{−} 5  4 2 42 6 4 33 222 42
S_{4} × C_{2}  2  0 -3 -1  22 6 42 6 4 4 42 33
46.  2_{−}  4_{+} 9_{−} -1  5  0 1^{5} 4_{+} ∗  1^{5} 9_{−} 5  6 222 33 6 4 5 222 33
S_{5}  -5 16 -1  22 222 22 4 4 222 5 5
47.  2_{−}  10_{+} 5_{−}  3  0 10 2^{3} 10_{+} ∗  1^{5} 5_{−} 5  6 4 33 4 4 5 4 33
S_{5}  0 -24 23  33 4 5 4 6 6 5 22
48.  2_{−}  6_{+} 9_{−}  3  0 -25 2^{3} 6_{+} ∗  1^{5} 9_{−} 5  4 4 5 222 4 5 4 33
S_{5}  0 165 170  22 222 22 4 6 6 33 5
49.  2_{−}  10_{+} 7_{−}  0 -15  0 2^{3} 10_{+} ∗  1^{5} 7_{−} 5  4 4 33 6 4 22 4 5
S_{5}  75 135 -50  22 4 5 4 222 6 33 33
50.  2_{−}  10_{+} 7_{−} -3  0  5 2^{3} 10_{+} ∗  1^{5} 7_{−} 5  6 222 5 4 4 5 4 33
S_{5}  225 24 -187  22 6 33 4 4 4 5 22
51.  2_{−}  10_{+} 7_{−}  0 -15 -30 2^{3} 10_{+} ∗  1^{5} 7_{−} 5  4 222 5 6 6 5 222 33
S_{5}  -15 -9 -20  33 6 5 4 4 4 33 22
52.  2_{−}  8_{+} 9_{−}  0 15 -20 2^{3} 8_{+} ∗  1^{5} 9_{−} 5  6 4 5 6 6 33 6 5
S_{5}  3  30 -168 -145  22 6 22 6 4 6 5 5
53.  2_{−}  8_{+} 9_{−}  3 -15 -35 2^{3} 8_{+} ∗  1^{5} 9_{−} 5  4 222 22 4 6 22 4 22
S_{5}  15 168 239  33 222 5 6 4 6 5 5
54.  2_{−}  8_{+} 9_{−}  0 -15 -10 2^{3} 8_{+} ∗  1^{5} 9_{−} 5  222 4 5 4 6 5 4 5
S_{5}  -45 -6 -380  33 4 5 4 4 4 33 22
55.  2_{−}  10_{+} 9_{−}  3  0 -35 2^{3} 10_{+} ∗  1^{5} 9_{−} 5  4 222 5 4 4 33 4 5
S_{5}  -90 -339 -442  5 4 5 4 6 6 22 22
56.  2_{−}  10_{+} 9_{−} -3 45 -70 2^{3} 10_{+} ∗  1^{5} 9_{−} 5  6 4 22 4 222 33 4 5
S_{5}  675 -534 77  33 6 22 6 4 4 5 22
57.  2_{−}  10_{+} 9_{−}  0 -15 15 2^{3} 10_{+} ∗  1^{5} 9_{−} 5  6 4 33 222 6 5 4 5
S_{5}  30 -441 -155  5 222  6 222 6 5 33
58.  2_{−}  10_{+} 9_{−}  3  0 -110 2^{3} 10_{+} ∗  1^{5} 9_{−} 5  6 4 5 4 222 22 4 22
S_{5}  -315 -609 -127  22 6 33 6 222 6 5 5
59.  2_{−}  10_{+} 9_{−}  0 -15 -60 2^{3} 10_{+} ∗  1^{5} 9_{−} 5  222 6 33 6 6 5 4 22
S_{5}  -195 -396 -80  22 222 33 6 6 6 5 33