Old Sextic Fields Unramified Outside of {∞, 3, 11}

Discriminant class: −11

∞  3 11 a_{1} a_{2} a_{3}  Over p = 3  Over p = 11  2 5 7 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}  8_{+} 3_{k} 0 -6 0 1^{3} 4_{+} 1  1^{2} 1_{i} 11∗ 11∗ 222 33 222 222 222 222  222
S_{3}  3  9 0 44 1^{3} 4_{+} 1  1^{2} 1_{i} 11∗ 11∗ 33 33 222 222 33 33 33 222
2.  3_{i}  8_{+} 3_{k} -3 6 -5 1^{3} 4_{+} 1  3^{2} 3_{k} 11∗  6 33 6 6 222 222 33 6
C_{6}  3  33 -54 111 1^{3} 4_{+} 1  33  6 6 33  33 6
3.  3_{i}  10_{−} 3_{i} -3 12 -25 1^{3} 5_{i} 3∗  2^{2} 2_{+} 1  V 222 22 6 6 222 6 22 222
D_{6}  45 18 12 1^{3} 5_{i} 3∗  1^{2} 1_{i} 11∗ 11∗ 33 33 222 6 22 22 22 222
4.  3_{i}  10_{−} 3_{i} 0 0 3 1^{3} 5_{i} 3∗  2^{2} 2_{+} 1  V 6 22 222 222 6 222 22 6
D_{6}  2  0 0 27 1^{3} 5_{i} 3∗  1^{2} 1_{i} 11∗ 11∗  33 6 222 22 22 22 222
5.  3_{i}  6_{−} 5_{i} 0 0 0 1^{3} 3_{i} 3∗  1^{6} 5_{i} 11∗  222 22 6 6 222 222 22 222
D_{6}  2  0 0 11 1^{3} 3_{i} 3∗  33  222 222 22 22 22 222
6.  3_{i}  10_{−} 5_{i} 0 0 0 1^{3} 5_{i} 3∗  1^{6} 5_{i} 11∗  222 22 222 222 222 6 22 222
D_{6}  0 0 99 1^{3} 5_{i} 3∗  33 33 222 6 22 22 22 222
7.  3_{i}  10_{−} 5_{i} 0 0 0 1^{3} 5_{i} 3∗  1^{6} 5_{i} 11∗  222 22 6 6 222 6 22 222
D_{6}  3  0 0 891 1^{3} 5_{i} 3∗  33 222 6 22 22 22 222
8.  3_{i}  4_{+} 3_{k} 0 -1 -2 1^{3} 4_{+} 1  3^{2} 3_{k} 11∗  6 3 6 6 222 222 33 6
S_{3} × C_{3}  3 1 1 3 0_{+} 1  3 33 6 6 3 33 3 6
9.  3_{i}  8_{+} 3_{k} 3 6 7 1^{3} 4_{+} 1  1^{4} 3_{k} 11 11∗ 4 33 4 222 222 222 22 4
S_{4}^{−}  3  -6 -9 6 1^{3} 4_{+} 1  2 0_{+} ∗  ∗ 33 33 222 222 33 33 33 4
10.  3_{i}  8_{−} 3_{i} 0 0 -1 1^{3} 5_{i} 3∗  2^{2} 2_{+} 1  V 6 22 6 6 6 6 22 6
S_{3} × S_{3}  0 0 3 1^{3} 3_{i} 3∗  1^{2} 1_{i} 11∗ 11∗ 33 3 6 6 22 22 22 222
11.  3_{i}  6_{−} 5_{i} 0 0 -11 1^{3} 5_{i} 3∗  1^{6} 5_{i} 11∗  6 22 6 6 6 6 22 6
S_{3} × S_{3}  0 0 33 1^{2} 1_{i} 3∗ 3∗  3 6 6 22 22 22 222
12.  3_{i}  8_{−} 5_{i} 0 0 -11 1^{3} 5_{i} 3∗  1^{6} 5_{i} 11∗  6 22 222 222 6 6 22 6
S_{3} × S_{3}  0 0 99 1^{3} 3_{i} 3∗  33 3 6 6 22 22 22 222
13.  3_{i}  10_{−} 5_{i} 0 0 -33 1^{3} 5_{i} 3∗  1^{6} 5_{i} 11∗  6 22 6 6 6 222 22 6
S_{3} × S_{3}  0 0 297 1^{3} 5_{i} 3∗  33 33 6 222 22 22 22 222
14.  1_{k}  10_{−} 3_{k} -3 -6 17 1^{3} 5_{i} 3∗  1^{2} 1_{k} 11 11 6 42 4 222 6 4 42 6
S_{4} × C_{2}  -9 0 9 1^{3} 5_{i} 3∗  1^{2} 1_{k} 11 11 22 33 6 4 42 22 22 4
15.  3_{i}  6_{−} 5_{i} -3 12 -30 1^{3} 3_{i} 3∗  1^{6} 5_{i} 11∗  4 42 6 6 4 2 42 222
S_{4} × C_{2}  2  36 -27 27 1^{3} 3_{i} 3∗  33 22 4 2 42 22 42 2

Discriminant class: −3

∞  3 11 a_{1} a_{2} a_{3}  Over p = 3  Over p = 11  2 5 7 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
16.  3_{i}  7_{k} 4_{+} -3  6 15 1^{6} 7_{k} 3∗  2^{3} 4_{+} ∗  222 222 33 33 222  222 222
S_{3}  4  -27 -36 144  33  222  222 222 222
17.  3_{i}  11_{k} 4_{+} 3  6 -59 1^{6} 11_{k} 3∗  2^{3} 4_{+} ∗  222 222 33 33 222 33 222 222
S_{3}  -93 102 1156  33 222 33 222 222 222
18.  3_{i}  11_{k} 4_{+} 0  0  0 1^{6} 11_{k} 3∗  2^{3} 4_{+} ∗  222 222  222 33 222 222
S_{3}  0  0 363  33 33 222 33 222 222 222
19.  3_{i}  9_{k} 0_{+} 0  0  -1 1^{6} 9_{k} 3∗  6 0_{+} ∗  6 6 33 33 222  6 6
C_{6}  0  0  1  33  6 33 6 222 6 33

Discriminant class: −3 (continued)

∞  3 11 a_{1} a_{2} a_{3}  Over p = 3  Over p = 11  2 5 7 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} 0_{+} 0  0  0 1^{6} 11_{k} 3∗  2 0_{+} ∗  ∗ 222 222 33 33 222 33 222 222
S_{3}  0  0  3  2 0_{+} ∗  ∗ 33 33 222 33 222 222 222
21.  3_{i}  9_{k} 2_{+} 0  6  0 1^{6} 9_{k} 3∗  2^{2} 2_{+} 1  V 222 6 22 22 222 22 222 222
D_{6}  3  9  0 48  2 0_{+} ∗  ∗ 33 33 222 22 6 6 6 22
22.  3_{i}  11_{k} 2_{+} 0 -9  0 1^{6} 11_{k} 3∗  2^{2} 2_{+} 1  V 6 222 22 22 6 22 222 6
D_{6}  2  81 -135 75  2 0_{+} ∗  ∗  33 6 22 222 222 222 22
23.  3_{i}  9_{k} 4_{+} 0  0 11 1^{6} 9_{k} 3∗  2^{3} 4_{+} ∗  6 6 3 3 222  6 6
S_{3} × C_{3}  0  0 121  3  6 33 6 222 6 33
24.  3_{i}  11_{k} 4_{+} 0  0 33 1^{6} 11_{k} 3∗  2^{3} 4_{+} ∗  6 6 3 3 222 33 6 6
S_{3} × C_{3}  0  0 1089  33 33 6 3 6 222 6 33
25.  3_{i}  11_{k} 4_{+} 0  0 -33 1^{6} 11_{k} 3∗  2^{3} 4_{+} ∗  6 6 33 33 222 33 6 6
S_{3} × C_{3}  0  0 363  3 33 6 3 6 222 6 33
26.  3_{i}  11_{k} 0_{+} 0  0  3 1^{6} 11_{k} 3∗  6 0_{+} ∗  6 6 3 3 222 33 6 6
S_{3} × C_{3}  0  0  3  3 33 6 3 6 222 6 33
27.  3_{i}  7_{k} 4_{+} 3  6 18 1^{6} 7_{k} 3∗  2^{3} 4_{+} ∗  4 4 33 33 4 22 4 222
S_{4}^{−}  8  39 36 12  33 22 4 22 4 222 4 22
28.  3_{i}  11_{k} 2_{+} 0  9 -12 1^{6} 11_{k} 3∗  2^{2} 2_{+} 1  V 6 6 22 22 6 22 222 6
S_{3} × S_{3}  27 -45 39  2 0_{+} ∗  ∗ 33 3 6 22 6 6 6 22
29.  1_{k}  11_{k} 2_{−} 0 -9  0 1^{6} 11_{k} 3∗  2 0_{+} ∗  ∗ 6 4 42 22 6 42 4 6
S_{4} × C_{2}  18  0  3  1^{2} 1_{k} 11 11 22 33 6 42 4 222 222 42
30.  3_{i}  9_{k} 4_{+} 3 15 33 1^{6} 9_{k} 3∗  1^{4} 3_{k} 11 11∗ 4 6 42 22 222 22 2 4
S_{4} × C_{2} 24  63 90 48  1^{2} 1_{i} 11∗ 11∗ 33 33 222 22 6 6 6 42

Discriminant class: 1

∞  3 11 a_{1} a_{2} a_{3}  Over p = 3  Over p = 11  2 5 7 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
31. 2_{−} 10_{−} 2_{+} -3 3 -1 1^{3} 5_{i} 3∗  2^{2} 2_{+} 1 V 33 42 42 22 33 42 42 33
S_{4}^{+}  -9 9 -3 1^{3} 5_{i} 3∗  22 33 33 42 42 22 22 42
32. 2_{−} 6_{−} 4_{+} 3 9 13 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  42 42 33 33 42 22 42 22
S_{4}^{+}  18 12 -3 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  33 22 42 22 42 22 42 22
33. 2_{−} 8_{+} 4_{−} 0 3 25 1^{3} 4_{+} 1  1^{4} 3_{k} 11 11∗ 42 33 42 22 22 22 22 42
S_{4}^{+}  3  27 -12 -17 1^{3} 4_{+} 1  1^{2} 1_{k} 11 11 33 33 22 22 33 33 33 42
34. 2_{−} 10_{−} 4_{+} 0 -3 -24 1^{3} 5_{i} 3∗  1^{5} 4_{+} 1  33 5 33 5 22 33 5 22
A_{5}  2  84 -90 29 1^{3} 5_{i} 3∗  5 5 5 22 33 33 5 22
35. 2_{−} 10_{−} 4_{+} 0 -3 -9 1^{3} 5_{i} 3∗  1^{5} 4_{+} 1  5 33 22 5 5 33 5 33
A_{5}  2  -15 -9 -4 1^{3} 5_{i} 3∗  33 5 33 5 5 5 5 22

Discriminant class: 33

∞  3 11 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 11  2 5 7 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
36.  0_{+}  9_{k} 3_{i}  0 -18  -8 1^{6} 9_{k} 3∗  3^{2} 3_{i} 11  33 6 6 6  222 6 33
C_{6}  81 72 -17  33  33 6 6 222 6 6
37.  0_{+}  11_{k} 3_{i}  0 -18  0 1^{6} 11_{k} 3∗  1^{2} 1_{k} 11 11 33 222 222 222 33 222 222 33
S_{3}  81 0 -33  1^{2} 1_{k} 11 11  33 33 222 222 222 222 222

Discriminant class: 33 (continued)

∞  3 11 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 11  2 5 7 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_{−}  9_{k} 3_{k} 0 0  1 1^{6} 9_{k} 3∗  2^{2} 2_{+} 1 V 22 6 222 222 22 222 222 22
D_{6}  3  0 0  -8  1^{2} 1_{k} 11 11 33 33 22 222 6 6 6 222
39.  2_{−}  11_{k} 3_{k} 0 0 -27 1^{6} 11_{k} 3∗  2^{2} 2_{+} 1 V 22 222 6 6 22 6 222 22
D_{6}  0 0 -24  1^{2} 1_{k} 11 11 33 33 22 6 222 222 222 222
40.  2_{−}  7_{k} 5_{k} 0 0  0 1^{6} 7_{k} 3∗  1^{6} 5_{k} 11  22 222 6 6 22 222 222 22
D_{6}  2  0 0 -297  33  22 222 222 222 222 222
41.  2_{−}  11_{k} 5_{k} 0 0  0 1^{6} 11_{k} 3∗  1^{6} 5_{k} 11  22 222 222 222 22 6 222 22
D_{6}  0 0 -2673  33 33 22 6 222 222 222 222
42.  2_{−}  11_{k} 5_{k} 0 0  0 1^{6} 11_{k} 3∗  1^{6} 5_{k} 11  22 222 6 6 22 6 222 22
D_{6}  3  0 0 -33  33 22 6 222 222 222 222
43.  0_{+}  11_{k} 3_{i}  0 -18  -3 1^{6} 11_{k} 3∗  3^{2} 3_{i} 11  3 6 6 6 33 222 6 3
S_{3} × C_{3}  81 27 -72  33 33 3 6 6 222 6 6
44.  2_{−}  11_{k} 3_{k} -3 -3  23 1^{6} 11_{k} 3∗  2^{2} 2_{+} 1 V 33 4 4 222 33 4 4 33
S_{4}^{−}  4  -39 33 -11  1^{2} 1_{k} 11 11 22 33 33 4 4 222 222 4
45.  2_{−}  11_{k} 3_{k} 0 0 -12 1^{6} 11_{k} 3∗  2^{2} 2_{+} 1 V 22 6 6 6 22 6 222 22
S_{3} × S_{3}  0 0  3  1^{2} 1_{k} 11 11 3 3 22 6 6 6 6 222
46.  2_{−}  9_{k} 5_{k} 0 0 -22 1^{6} 9_{k} 3∗  1^{6} 5_{k} 11  22 6 6 6 22 222 222 22
S_{3} × S_{3}  0 0 -11  3 33 22 222 6 6 6 222
47.  2_{−}  11_{k} 5_{k} 0 0 -33 1^{6} 11_{k} 3∗  1^{6} 5_{k} 11  22 6 6 6 22 6 222 22
S_{3} × S_{3}  3  0 0 264  33 3 22 6 6 6 6 222
48.  2_{−}  11_{k} 5_{k} 0 0 -66 1^{6} 11_{k} 3∗  1^{6} 5_{k} 11  22 6 222 222 22 6 222 22
S_{3} × S_{3}  0 0 -99  3 3 22 6 6 6 6 222
49.  2_{−}  9_{k} 3_{k} 3 3  1 1^{6} 9_{k} 3∗  1^{4} 3_{k} 11 11∗ 42 6 4 222 22 222 2 42
S_{4} × C_{2}  3  6 6  1  33 33 22 222 6 6 6 4
50.  2_{−}  7_{k} 5_{k} 0 0 -11 1^{6} 7_{k} 3∗  1^{6} 5_{k} 11  42 4 6 6 42 2 4 22
S_{4} × C_{2}  2  -33 -33 -11  33 22 42 2 4 222 4 2