Old Sextic Fields Unramified Outside of {∞, 2, 11}

Discriminant class: −22

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 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}  11_{+} 3_{k}  0  6 0 1^{6} 11_{+} 2∗  2^{2} 2_{+} 1  V 6 6 222 22 222 22 33 22
D_{6}  2  20  0 22  1^{2} 1_{k} 11 11 33 6 222 22  222 6 22
2.  3_{i}  11_{+} 5_{k}  0  0 0 1^{4} 8_{−} 1 {−2∗ \atop −1} 1^{6} 5_{k} 11  222 6 222 33 6 22 22 33
D_{6}  4  33  0 88 1^{2} 3_{−} 2∗ 2∗  22 6 6 22 22 6 222
3.  3_{i}  11_{+} 3_{k}  0  8 0 1^{6} 11_{+} 2∗  1^{4} 3_{k} 11 11∗ 6 6 222 42 222 42 33 22
S_{4} × C_{2}  22  0 22  33 6 4 42  2 6 22
4.  3_{i}  11_{−} 3_{i}  0 -2 0 1^{6} 11_{−} 2∗  1^{2} 1_{k} 11 11 6 6 4 42 4 42 33 22
S_{4} × C_{2}  2  -10  0 22  1^{2} 1_{i} 11∗ 11∗ 33 6 4 22 22 2 6 22
5.  3_{i}  11_{+} 3_{k}  0 -4 0 1^{6} 11_{+} 2∗  1^{4} 3_{k} 11 11∗ 6 6 4 22 4 22 33 22
S_{4} × C_{2}  0  0 22  33 6 222 42 22 2 6 22
6.  3_{i}  7_{+} 5_{k}  0  0 0 1^{2} 2_{i} −1 −1 1^{6} 5_{k} 11  4 6 4 33 6 42 42 33
S_{4} × C_{2}  2  -11  0 22 1^{2} 2_{i} −1 −1  42 6 6 22 42 6 222 22
7.  3_{i}  9_{+} 5_{k}  0 11 0 1^{4} 9_{+} 2∗ −1 1^{6} 5_{k} 11  222 6 4 33 6 42 22 33
S_{4} × C_{2}  2  44  0 88  42 6 6 42 22 6 4 22
8.  3_{i}  9_{+} 5_{k} -1 16 -28 1^{4} 9_{+} 2∗ −1 1^{6} 5_{k} 11  4 6 222 33 6 22 42 33
S_{4} × C_{2}  2  108 -148 64  22 6 6 42 42 6 4
9.  3_{i}  11_{−} 3_{i}  1  2 4 1^{4} 11_{−} 2∗ 2∗ 1^{4} 3_{i} 11∗ 11 6 6 4 5 6 33 5 5
S_{5}  4 -4 8  2 0_{+} ∗  ∗ 33 4 6 33 5 6 4 5

Discriminant class: −11

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 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
10.  3_{i}  4_{+} 3_{k} -1 2 -3 2^{3} 4_{+} ∗  1^{2} 1_{i} 11∗ 11∗ 33 33 222 222 222 222 33 222
S_{3}  2 -1 1  1^{2} 1_{i} 11∗ 11∗ 33 33 222 222  33 222
11.  3_{i}  4_{−} 5_{i} -1 5 -6 2^{2} 4_{−} 1 {−1 \atop −∗} 1^{6} 5_{i} 11∗  22 33 222 6 6 222 22 6
D_{6}  9 -5 9 2 0_{+} ∗ ∗  22 33 6 222 22 33 22 222
12.  3_{i}  6_{+} 3_{k} -2 2 -4 1^{6} 6_{+} ∗  1^{4} 3_{k} 11 11∗ 33 33 4 222 4 222 33 222
S_{4}^{−}  4 -2 2  2 0_{+} ∗  ∗ 33 33 222 4 22 22 33 222
13.  3_{i}  10_{+} 3_{k} 0 3 0 1^{6} 10_{+} ∗  1^{4} 3_{k} 11 11∗ 33 33 222 4 222 4 33 222
S_{4}^{−}  11 0 11  2 0_{+} ∗  ∗ 33 33 4 4  22 33 222
14.  3_{i}  10_{−} 3_{i} 0 7 0 1^{6} 10_{−} ∗  2^{2} 2_{+} 1  V 33 33 4 4 4 4 33 222
S_{4}^{−}  2  15 0 11  1^{2} 1_{i} 11∗ 11∗ 33 33 4 222 22 22 33 222
15.  3_{i}  8_{−} 5_{i} 0 11 0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{6} 5_{i} 11∗  42 33 4 6 6 4 42 6
S_{4} × C_{2}  2  33 0 11 2 0_{+} ∗ ∗  42 33 6 222 42 33 22 2
16.  3_{i}  12_{−} 5_{i} 0 0 0 1^{4} 9_{−} 2∗ −1 1^{6} 5_{i} 11∗  42 33 222 6 6 222 42 6
S_{4} × C_{2}  4  22 0 44 1^{2} 3_{+} 2 2  22 33 6 4 42 33 42 222
17.  3_{i}  12_{−} 5_{i} 0 0 0 1^{4} 9_{−} 2∗ −1 1^{6} 5_{i} 11∗  22 33 4 6 6 4 22 6
S_{4} × C_{2}  4  22 0 396 1^{2} 3_{+} 2 2  42 33 6 4 22 33 42 2

Discriminant class: −2

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 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
18.  3_{i}  11_{i} 2_{−} 0 -2 0 1^{6} 11_{i} −2  1^{2} 1_{i} 11∗ 11∗ 33 6 222 222 22 22 6 222
D_{6}  0 0 2  1^{2} 1_{i} 11∗ 11∗ 6 6 22 22 222 222 33 222
19.  3_{i}  11_{k} 4_{+} 0 8 0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{3} 2_{+} ∗  22 6 222 6 33 22 222 6
D_{6}  2  25 0 32 1^{2} 3_{i} −2 −2 1^{3} 2_{+} ∗  222 6 33 22 222 6 22 222
20.  3_{i}  11_{k} 2_{+} 0 2 0 1^{6} 11_{k} −2  2^{2} 2_{+} 1  V 33 6 4 4 42 42 6 222
S_{4} × C_{2}  2 0 2  6 6 42 22 2 2 33 222
21.  3_{i}  7_{k} 4_{+} 2 8 10 2^{2} 4_{−} 1 {−1 \atop −∗} 1^{3} 2_{+} ∗  42 6 4 6 33 42 4 6
S_{4} × C_{2}  8 2 1 1^{2} 3_{i} −2 −2 1^{3} 2_{+} ∗  4 6 33 22 4 6 22 2
22.  3_{i}  11_{k} 4_{+} 0 7 0 1^{4} 9_{−} 2∗ −1 1^{3} 2_{+} ∗  42 6 222 6 33 22 4 6
S_{4} × C_{2}  2  20 0 18 1^{2} 2_{i} −∗ −∗ 1^{3} 2_{+} ∗  222 6 33 42 4 6 42 222
23.  3_{i}  11_{i} 4_{−} 0 0 0 1^{6} 11_{i} −2  1^{4} 3_{k} 11 11∗ 33 6 222 4 22 42 6 222
S_{4} × C_{2}  2  -22 0 242  1^{2} 1_{k} 11 11 6 6 42 42 222 2 33 222
24.  3_{i}  11_{i} 4_{−} 0 0 0 1^{6} 11_{i} −2  1^{4} 3_{k} 11 11∗ 33 6 4 222 42 22 6 222
S_{4} × C_{2}  2  44 0 242  1^{2} 1_{k} 11 11 6 6 22 42 2 2 33 222
25.  3_{i}  11_{k} 4_{+} 0 -1 0 1^{4} 9_{−} 2∗ −1 1^{3} 2_{+} ∗  22 6 4 6 33 42 222 6
S_{4} × C_{2}  2  4 0 2 1^{2} 2_{i} −∗ −∗ 1^{3} 2_{+} ∗  4 6 33 42 222 6 42 2

Discriminant class: −1

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 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
26.  3_{i}  6_{k} 4_{+} 0 -7 0 1^{2} 2_{i} −1 −1 2^{3} 4_{+} ∗  222 33 222 33 33 222 222 33
S_{3}  20 0 4 1^{2} 2_{i} −1 −1  222 33 33 222 222 33 222
27.  3_{i}  8_{k} 2_{+} -2 2 -2 1^{6} 8_{k} −1  2^{2} 2_{+} 1  V 6 33 222 22 22 222 6 22
D_{6}  4 -4 2  2 0_{+} ∗  ∗ 6 33 22 222 222  6 22
28.  3_{i}  10_{k} 4_{+} 0 4 0 1^{4} 8_{−} 1 {−2∗ \atop −1} 2^{3} 4_{+} ∗  4 33 4 33 33 4 4 33
S_{4}^{−}  4  9 0 4 1^{2} 2_{i} −1 −1  4 33 33 222 4 33 222 22
29.  3_{i}  12_{k} 4_{+} 0 4 0 1^{4} 9_{+} 2∗ −1 2^{3} 4_{+} ∗  222 33 4 33 33 4 222 33
S_{4}^{−}  2  -2 0 4 1^{2} 3_{k} −2∗ −2∗  4 33 33 4 222 33 4 22
30.  3_{i}  12_{k} 4_{+} 0 -2 0 1^{4} 9_{+} 2∗ −1 2^{3} 4_{+} ∗  4 33 222 33 33 222 4 33
S_{4}^{−}  2  -6 0 16 1^{2} 3_{k} −2∗ −2∗  222 33 33 4 4 33 4
31.  3_{i}  10_{i} 2_{−} 0 -1 0 1^{6} 10_{i} −1  2 0_{+} ∗  ∗ 6 33 4 42 42 4 6 22
S_{4} × C_{2}  1 0 1  1^{2} 1_{i} 11∗ 11∗ 6 33 42 222 2 22 6 22
32.  3_{i}  8_{k} 4_{+} 2 3 14 1^{6} 8_{k} −1  1^{4} 3_{k} 11 11∗ 6 33 4 22 42 222 6 22
S_{4} × C_{2}  2  35 56 47  1^{2} 1_{i} 11∗ 11∗ 6 33 22 4 2 22 6 22
33.  3_{i}  10_{k} 4_{+} 0 11 0 1^{6} 10_{k} −1  1^{4} 3_{k} 11 11∗ 6 33 222 42 22 4 6 22
S_{4} × C_{2}  2  33 0 121  1^{2} 1_{i} 11∗ 11∗ 6 33 42 4 222 22 6 22

Discriminant class: 1

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 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
34. 2_{−} 10_{−} 2_{+} 0 1  0 1^{6} 10_{−} 1  2^{2} 2_{+} 1 V 33 33 42 42 42 42 33 22
S_{4}^{+}  1 0 -1  33 33 42 22 22 22 33 22

Discriminant class: 1 (continued)

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 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
35. 2_{−} 6_{+} 4_{−} 2 1 -6 1^{6} 6_{+} 1  1^{4} 3_{k} 11 11∗ 33 33 42 22 42 22 33 22
S_{4}^{+}  -17 -22 -13  1^{2} 1_{k} 11 11 33 33 22 42 22 22 33 22
36. 2_{−} 8_{−} 4_{+} 0 -4  0 1^{4} 8_{−} 1 {−2∗ \atop −1} 1^{3} 2_{+} ∗  42 33 42 33 33 42 42 33
S_{4}^{+}  9 0 -4  1^{3} 2_{+} ∗  42 33 33 22 42 33 22 22
37. 2_{−} 10_{+} 4_{−} 0 -11  0 1^{6} 10_{+} 1  1^{4} 3_{k} 11 11∗ 33 33 22 42 22 42 33 22
S_{4}^{+}  33 0 -121  1^{2} 1_{k} 11 11 33 33 42 42  22 33 22
38. 2_{−} 12_{−} 4_{+} 0 2  0 1^{4} 9_{+} 2∗ −1 1^{3} 2_{+} ∗  42 33 22 33 33 22 42 33
S_{4}^{+}  2  -6 0 -16 1^{2} 3_{−} 2∗ 2∗ 1^{3} 2_{+} ∗  22 33 33 42 42 33 42
39. 2_{−} 12_{−} 4_{+} 0 -4  0 1^{4} 9_{+} 2∗ −1 1^{3} 2_{+} ∗  22 33 42 33 33 42 22 33
S_{4}^{+}  2  -2 0 -4 1^{2} 3_{−} 2∗ 2∗ 1^{3} 2_{+} ∗  42 33 33 42 22 33 42 22

Discriminant class: 2

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 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
40.  2_{−}  11_{−} 2_{+} 0 2  0 1^{6} 11_{−} 2  2^{2} 2_{+} 1  V 6 6 22 222 22 222 33 222
D_{6}  0 0 -2  2 0_{+} ∗  ∗ 33 6 22 222  222 6 222
41.  2_{−}  11_{−} 4_{+} 0 -8  0 1^{4} 8_{−} 1 {−2 \atop −1} 2^{3} 4_{+} ∗  222 6 22 6 33 222 22 6
D_{6}  2  25 0 -32 1^{2} 3_{+} 2 2  22 6 33 222 22 6 222 222
42.  2_{−}  11_{+} 2_{−} 0 -2  0 1^{6} 11_{+} 2  2 0_{+} ∗  ∗ 6 6 42 4 42 4 33 222
S_{4} × C_{2}  2 0 -2  1^{2} 1_{i} 11∗ 11∗ 33 6 42 222 22 2 6 222
43.  2_{−}  7_{−} 4_{+} 2 2  2 2^{2} 4_{−} 1 {−1 \atop −∗} 2^{3} 4_{+} ∗  4 6 42 6 33 4 42 6
S_{4} × C_{2}  -2 2 -1 1^{2} 3_{+} 2 2  42 6 33 222 42 6 222 2
44.  2_{−}  9_{−} 4_{+} 0 1  0 1^{4} 9_{−} 2∗ −1 2^{3} 4_{+} ∗  222 6 42 6 33 4 22 6
S_{4} × C_{2}  4 0 -2 2 0_{+} ∗ ∗  42 6 33 4 22 6 4 2
45.  2_{−}  9_{−} 4_{+} 0 -7  0 1^{4} 9_{−} 2∗ −1 2^{3} 4_{+} ∗  4 6 22 6 33 222 42 6
S_{4} × C_{2}  20 0 -18 2 0_{+} ∗ ∗  22 6 33 4 42 6 4 222
46.  2_{−}  11_{−} 4_{+} 0 0  0 1^{6} 11_{−} 2  1^{4} 3_{k} 11 11∗ 6 6 22 4 22 4 33 222
S_{4} × C_{2}  -22 0 -242  1^{2} 1_{i} 11∗ 11∗ 33 6 42 4  2 6 222
47.  2_{−}  11_{−} 4_{+} 0 0  0 1^{6} 11_{−} 2  1^{4} 3_{k} 11 11∗ 6 6 42 222 42 222 33 222
S_{4} × C_{2}  44 0 -242  1^{2} 1_{i} 11∗ 11∗ 33 6 22 4 22 2 6 222

Discriminant class: 11

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 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
48.  2_{−}  8_{k} 3_{k} -2 0  6 1^{6} 8_{k} −∗  2^{2} 2_{+} 1  V 6 33 22 222 222 22 6 222
D_{6}  -2 -8 -2  1^{2} 1_{k} 11 11 6 33 222 22 222  6 222
49.  2_{−}  6_{k} 5_{k} -2 -2 -4 2^{2} 4_{−} 1 {−1 \atop −∗} 1^{6} 5_{k} 11  222 33 22 6 6 22 222 6
D_{6}  -10 16 -7 1^{2} 2_{i} −∗ −∗  222 33 6 22 222 33 222 222
50.  2_{−}  8_{k} 5_{k} -2 -2 -4 1^{6} 8_{k} −∗  1^{6} 5_{k} 11  6 3 22 6 6 22 6 6
S_{3} × S_{3}  12 16  4  6 3 6 22 222 33 6 222
51.  2_{−}  8_{k} 3_{k} 2 5 10 1^{6} 8_{k} −∗  1^{4} 3_{k} 11 11∗ 6 33 42 222 4 22 6 222
S_{4} × C_{2}  5 0 -1  6 33 222 42 2 22 6 222

Discriminant class: 11 (continued)

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 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
52.  2_{−}  10_{i} 3_{i}  0 -7  0 1^{6} 10_{i} −∗  1^{2} 1_{i} 11∗ 11∗ 6 33 42 4 4 42 6 222
S_{4} × C_{2}  15 0 -11  1^{2} 1_{k} 11 11 6 33 4 22 2 22 6 222
53.  2_{−}  10_{k} 3_{k} 0 -3  0 1^{6} 10_{k} −∗  1^{4} 3_{k} 11 11∗ 6 33 22 4 222 42 6 222
S_{4} × C_{2}  11 0 -11  6 33 4 42 222 22 6 222
54.  2_{−}  10_{k} 5_{k} 0 -11  0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{6} 5_{k} 11  4 33 42 6 6 42 4 6
S_{4} × C_{2}  2  33 0 -11 1^{2} 2_{i} −∗ −∗  4 33 6 22 4 33 222 2
55.  2_{−}  12_{k} 5_{k} 0 0  0 1^{4} 9_{−} 2∗ −1 1^{6} 5_{k} 11  4 33 22 6 6 22 4 6
S_{4} × C_{2}  2  22 0 -44 1^{2} 3_{i} −2 −2  222 33 6 42 4 33 4 222
56.  2_{−}  12_{k} 5_{k} 0 0  0 1^{4} 9_{−} 2∗ −1 1^{6} 5_{k} 11  222 33 42 6 6 42 222 6
S_{4} × C_{2}  2  22 0 -396 1^{2} 3_{i} −2 −2  4 33 6 42 222 33 4 2

Discriminant class: 22

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 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
57.  2_{−}  11_{k} 3_{k} 0 -6 0 1^{6} 11_{k} −2∗  1^{2} 1_{i} 11∗ 11∗ 33 6 22 22 222 222 6 22
D_{6}  20 0 -22  1^{2} 1_{i} 11∗ 11∗ 6 6 222 222 222 222 33 22
58.  2_{−}  11_{i} 5_{i}  0 0 0 1^{4} 8_{−} 1 {−2∗ \atop −1} 1^{6} 5_{i} 11∗  22 6 22 33 6 222 222 33
D_{6}  2  33 0 -88 1^{2} 3_{k} −2∗ −2∗  222 6 6 222 222 6 22
59.  2_{−}  11_{k} 3_{k} 0 -8 0 1^{6} 11_{k} −2∗  1^{4} 3_{k} 11 11∗ 33 6 22 42 222 4 6 22
S_{4} × C_{2}  22 0 -22  2 0_{+} ∗  ∗ 6 6 4 4 222 2 33 22
60.  2_{−}  11_{k} 3_{k} 0 4 0 1^{6} 11_{k} −2∗  1^{4} 3_{k} 11 11∗ 33 6 42 22 4 222 6 22
S_{4} × C_{2}  0 0 -22  2 0_{+} ∗  ∗ 6 6 222 4 2 2 33 22
61.  2_{−}  11_{i} 3_{i}  0 2 0 1^{6} 11_{i} −2∗  2^{2} 2_{+} 1  V 33 6 42 42 4 4 6 22
S_{4} × C_{2}  -10 0 -22  1^{2} 1_{i} 11∗ 11∗ 6 6 4 222 2 2 33 22
62.  2_{−}  7_{i} 5_{i}  0 0 0 1^{2} 2_{i} −1 −1 1^{6} 5_{i} 11∗  42 6 42 33 6 4 4 33
S_{4} × C_{2}  -11 0 -22 1^{2} 2_{i} −1 −1  4 6 6 222 4 6 22 22
63.  2_{−}  11_{i} 5_{i}  0 -22 0 1^{4} 9_{+} 2∗ −1 1^{6} 5_{i} 11∗  42 6 22 33 6 222 4 33
S_{4} × C_{2}  2  66 0 -88 1^{2} 2_{i} −1 −1  222 6 6 4 4 6 42
64.  2_{−}  11_{i} 5_{i}  0 -11 0 1^{4} 9_{+} 2∗ −1 1^{6} 5_{i} 11∗  22 6 42 33 6 4 222 33
S_{4} × C_{2}  2  44 0 -88 1^{2} 2_{i} −1 −1  4 6 6 4 222 6 42 22
65.  2_{−}  11_{k} 3_{k} -1 -1 2 1^{4} 11_{k} −2 −2∗ 1^{4} 3_{k} 11 11∗ 5 6 5 33 4 6 222 33
S_{5}  -5 -1 -3 2 0_{+} ∗  ∗ 2 0_{+} ∗  ∗ 6 4 222 4 6 4 33 5