Old Sextic Fields Unramified Outside of {∞, 3, 17}

Discriminant class: −51

∞  3 17 a_{1} a_{2} a_{3}  Over p = 3  Over p = 17  2 5 7 11 13 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  9  0 1^{6} 7_{i} 3  1^{2} 1_{−} 17∗ 17∗ 222 33 222  33 33 33 33
S_{3}  2  -3 -27 33  1^{2} 1_{−} 17∗ 17∗ 222 222 33 33 222 222 222 222
2.  3_{i}  9_{i} 3_{−} 0 24 -13 1^{6} 9_{i} 3  1^{2} 1_{−} 17∗ 17∗ 6 33 6 33 33  33 33
C_{6}  8  144 -156 361  1^{2} 1_{−} 17∗ 17∗ 6 222 33 33 6 222 6 6
3.  3_{i}  3_{k} 5_{+} 3  8 11 2^{2} 2_{+} 1 V 1^{6} 5_{+} 17∗  222 22 6 22 33 33 22 22
D_{6}  2  19 14  4 1^{2} 1_{k} 3 3  6 6 22 33 222 222 222 6
4.  3_{i}  11_{k} 3_{+} 0  0 36 1^{6} 11_{k} 3  2^{2} 2_{−} 1  V 222 22 6 22 33 33 22 22
D_{6}  4  0  0 375  1^{2} 1_{−} 17∗ 17∗ 6 6 22 33 222 222 222 222
5.  3_{i}  11_{k} 5_{+} 0  0  0 1^{6} 11_{k} 3  1^{6} 5_{+} 17∗  222 22 6 22  33 22 22
D_{6}  6  0  0 51  6 222 22  222 222 222 6
6.  3_{i}  11_{k} 5_{+} 0  0  0 1^{6} 11_{k} 3  1^{6} 5_{+} 17∗  222 22 222 22 33  22 22
D_{6}  18  0  0 4131  222 6 22 33 222 222 222 6
7.  3_{i}  9_{i} 3_{−} 0  6  8 1^{6} 9_{i} 3  1^{2} 1_{−} 17∗ 17∗ 6 3 6 33 3 33 3 3
S_{3} × C_{3}  2  9 24 67  1^{2} 1_{−} 17∗ 17∗ 6 222 3 3 6 222 6 6
8.  1_{k}  9_{i} 1_{+} 0 -3  -5 1^{6} 9_{i} 3  1^{2} 1_{+} 17 17 6 33 6 33 33 22 33 33
A_{4} × C_{2}  0  3  1  2 0_{+} ∗  ∗ 6 222 33 33 6 2 6 6
9.  3_{i}  11_{k} 1_{+} 0  0  6 1^{6} 11_{k} 3  4 0_{+} ∗  ∗ 4 22 6 42 33 33 42 22
S_{4} × C_{2}  2  0 -9  6  1^{2} 1_{+} 17 17 6 6 42 33 4 222 222 222
10.  3_{i}  3_{k} 5_{+} -3  8 -11 1^{4} 3_{k} 3 3∗ 1^{6} 5_{+} 17∗  4 42 6 22 33 33 42 22
S_{4} × C_{2}  2  36 -31 72  6 6 22 33 222 222 222 6

Discriminant class: −3

∞  3 17 a_{1} a_{2} a_{3}  Over p = 3  Over p = 17  2 5 7 11 13 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
11.  3_{i}  3_{k} 4_{+} 0 1  0 1^{2} 1_{i} 3∗ 3∗ 2^{3} 4_{+} ∗  222 222 33 222 33 33 222 222
S_{3}  -11 0 12 1^{2} 1_{i} 3∗ 3∗  33 33 222 33 222 222 222 33
12.  3_{i}  11_{k} 4_{+} 0 0  0 1^{6} 11_{k} 3∗  2^{3} 4_{+} ∗  222 222  222 33  222 222
S_{3}  81  0 0 867  33 222 33 222 222 222 33
13.  3_{i}  11_{k} 4_{+} -3 6 -109 1^{6} 11_{k} 3∗  2^{3} 4_{+} ∗  222 222 33 222  33 222 222
S_{3}  9  159 150 2500  33  222  222 222 222 33
14.  3_{i}  9_{k} 0_{+} 0 0  -1 1^{6} 9_{k} 3∗  2 0_{+} ∗  ∗ 6 6 33 6 33  6 6
C_{6}  0 0  1  2 0_{+} ∗  ∗ 33  6 33 6 222 6 33
15.  3_{i}  11_{k} 0_{+} 0 0  0 1^{6} 11_{k} 3∗  2 0_{+} ∗  ∗ 222 222 33 222 33 33 222 222
S_{3}  0 0  3  2 0_{+} ∗  ∗ 33 33 222 33 222 222 222
16.  3_{i}  7_{i} 2_{−} 0 3  4 1^{6} 7_{i} 3∗  2^{2} 2_{−} 1  V 222 6 22 222 33 33 6 6
D_{6}  -3 -6  4  2 0_{+} ∗  ∗ 22 22 6 33 222 222 222 22
17.  3_{i}  9_{k} 4_{+} 0 0 -17 1^{6} 9_{k} 3∗  2^{3} 4_{+} ∗  6 6 3 6 3 33 6 6
S_{3} × C_{3}  9  0 0 289  3 33 6 3 6 222 6 3
18.  3_{i}  11_{k} 4_{+} 0 0 51 1^{6} 11_{k} 3∗  2^{3} 4_{+} ∗  6 6 33 6 3  6 6
S_{3} × C_{3}  9  0 0 867  33 33 6 3 6 222 6 3
19.  3_{i}  11_{k} 4_{+} 0 0 51 1^{6} 11_{k} 3∗  2^{3} 4_{+} ∗  6 6 3 6 33 33 6 6
S_{3} × C_{3}  9  0 0 2601  3  6 33 6 222 6 3

Discriminant class: −3 (continued)

∞  3 17 a_{1} a_{2} a_{3}  Over p = 3  Over p = 17  2 5 7 11 13 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  3 1^{6} 11_{k} 3∗  2 0_{+} ∗  ∗ 6 6 3 6 3 33 6 6
S_{3} × C_{3}  0 0  3  2 0_{+} ∗  ∗ 3 33 6 3 6 222 6 33
21.  3_{i}  3_{k} 4_{+} -3 10 -15 1^{4} 3_{k} 3 3∗ 2^{3} 4_{+} ∗  4 4 33 222 33 33 4 222
S_{4}^{−}  12 -5  2 2 0_{+} ∗ ∗  33 33 222 33 222 222 222 33
22.  1_{k}  9_{k} 2_{−} 0 -9 10 1^{6} 9_{k} 3∗  1^{2} 1_{−} 17∗ 17∗ 6 6 33 6 33 22 6 6
A_{4} × C_{2}  9 -27 19  1^{2} 1_{+} 17 17 33  6 33 6 2 6 33
23.  3_{i}  11_{k} 2_{+} 3 6  -8 1^{6} 11_{k} 3∗  2^{2} 2_{+} ∗  ∗ 4 222 33 4 33 33 4 222
S_{4}^{−}  4  -12 -15 73  33 33 4 33 4 222 222
24.  3_{i}  11_{k} 4_{+} 0 9 -12 1^{6} 11_{k} 3∗  1^{5} 4_{+} ∗  4 6 22 6 5 33 4 4
S_{5}  8  45 -63 48  5 5 222 22 4 4 6 5

Discriminant class: 1

∞  3 17 a_{1} a_{2} a_{3}  Over p = 3  Over p = 17 2 5 7 11 13 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
25. 2_{−} 8_{+} 2_{−} 0 0 7 1^{3} 4_{+} 1  2 0_{+} ∗ ∗ 33 33 33 33 33 22 33 33
A_{4} -6 9 -3 1^{3} 4_{+} 1 1^{2} 1_{+} 17 17 33 33 33 33 22 33 33
26. 2_{−} 4_{−} 4_{+} 3 3 1 1^{4} 3_{k} 3 3∗ 1^{3} 2_{+} ∗  42 42 33 22 33 33 42 22
S_{4}^{+} 0 0 -17 1^{2} 1_{k} 3 3 1^{3} 2_{+} ∗ 33 33 22 33 22 22 22 33
27. 2_{−} 10_{−} 2_{+} 0 6 15 1^{3} 5_{i} 3∗  2^{2} 2_{+} ∗ ∗ 42 22 33 42 33 33 42 22
S_{4}^{+}  -15 -18 -4 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 33 33 42 33 42 22 22
28. 2_{−} 10_{−} 4_{+} 3 -15 -35 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  5 33 22 5 5 22 33 33
A_{5}  2  90 261 192 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  5 5 33 22 5 5 33 5

Discriminant class: 17

∞  3 17 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 17  2 5 7 11 13 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
29.  0_{+}  8_{+} 3_{+} -3 -15  37 2^{3} 8_{+} ∗  1^{2} 1_{+} 17 17 33 6 6 6 33  6 6
C_{6}  33 -33  -1  1^{2} 1_{+} 17 17 6 222 6 33 33  33 6
30.  2_{−}  6_{+} 3_{−} 0 0  -8 2^{3} 6_{+} ∗  2^{2} 2_{−} 1  V 22 6 222 222 33 33 6 6
D_{6}  0 0  -1  1^{2} 1_{+} 17 17 222 222 6 33 22 22 22 222
31.  2_{−}  2_{+} 5_{−} 0 0  0 2^{2} 2_{+} 1 V 1^{6} 5_{−} 17  22 222 6 222 33 33 222 222
D_{6}  0 0 -17 2 0_{+} ∗ ∗  6 6 222 33 22 22 22 6
32.  2_{−}  10_{+} 3_{−} -3 -9  29 2^{3} 10_{+} ∗  2^{2} 2_{−} 1  V 22 222 6 222 33 33 222 222
D_{6}  2  27 33 -94  1^{2} 1_{+} 17 17 6 6 222 33 22 22 22 222
33.  2_{−}  10_{+} 5_{−} 0 0  0 2^{3} 10_{+} ∗  1^{6} 5_{−} 17  22 222 6 222  33 222 222
D_{6}  3  0 0 -1377  6 222 222  22 22 22 6
34.  2_{−}  10_{+} 5_{−} 0 0  0 2^{3} 10_{+} ∗  1^{6} 5_{−} 17  22 222 222 222 33  222 222
D_{6}  9  0 0 -153  222 6 222 33 22 22 22 6
35.  2_{−}  8_{+} 1_{−} -3 3  -3 2^{3} 8_{+} ∗  1^{2} 1_{−} 17∗ 17∗ 33 6 6 6 33 22 6 6
A_{4} × C_{2}  3 -3  1  2 0_{+} ∗  ∗ 6 222 6 33 33 22 33 6
36.  2_{−}  10_{+} 3_{−} 0 0 -24 2^{3} 10_{+} ∗  2^{2} 2_{−} 1  V 22 6 6 222 3 3 6 6
S_{3} × S_{3}  0 0  -9  1^{2} 1_{+} 17 17 6 6 6 3 22 22 22 222
37.  2_{−}  6_{+} 5_{−} 0 0 -34 2^{3} 6_{+} ∗  1^{6} 5_{−} 17  22 6 6 222 3 3 6 6
S_{3} × S_{3}  3  0 0  17  6 6 6 3 22 22 22 6

Discriminant class: 17 (continued)

∞  3 17 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 17  2 5 7 11 13 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_{+} 5_{−} 0 0 -306 2^{3} 10_{+} ∗  1^{6} 5_{−} 17  22 6 6 222 33 3 6 6
S_{3} × S_{3}  3  0 0 1377  6 222 6 33 22 22 22 6
39.  2_{−}  10_{+} 5_{−} 0 0 -102 2^{3} 10_{+} ∗  1^{6} 5_{−} 17  22 6 222 222 3 33 6 6
S_{3} × S_{3}  3  0 0 153  222 6 6 3 22 22 22 6
40.  2_{−}  10_{+} 1_{−} -3 0  8 2^{3} 10_{+} ∗  4 0_{+} ∗  ∗ 42 222 6 4 33 33 4 222
S_{4} × C_{2}  -9 6  -4  1^{2} 1_{−} 17∗ 17∗ 6 6 4 33 42 22 22 222
41.  2_{−}  4_{+} 5_{−} 0 0  17 1^{4} 3_{k} 3 3∗ 1^{6} 5_{−} 17  42 4 6 222 33 33 4 222
S_{4} × C_{2}  2  -51 51 -17 1^{2} 1_{i} 3∗ 3∗  6 6 222 33 22 22 22 6