Old Sextic Fields Unramified Outside of {∞, 11, 17}

Discriminant class: −187

∞  11 17 a_{1} a_{2} a_{3}  Over p = 11  Over p = 17  2 3 5 7 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}  5_{k} 3_{+} 2 -2 4 1^{6} 5_{k} 11  2^{2} 2_{−} 1  V 6 222 222 22 6 6 222 22
D_{6}  4  56 83 59  1^{2} 1_{−} 17∗ 17∗ 222 222 22 6 33  22
2.  3_{i}  3_{k} 5_{+} 0  0 -34 2^{2} 2_{+} 1 V 1^{6} 5_{+} 17∗  222 6 6 22 222 222 6 22
D_{6}  2  187 -374 476 1^{2} 1_{k} 11 11  222 6 22 222 33  33 22
3.  3_{i}  5_{k} 5_{+} 0  0 0 1^{6} 5_{k} 11  1^{6} 5_{+} 17∗  6 6 6 22 6 6 6 22
S_{3} × S_{3} 12  187 -374 187  222 6 22 6 3  33 22
4.  1_{k}  5_{k} 3_{−} -3 -10 -8 1^{6} 5_{k} 11  1^{2} 1_{+} 17 17 6 4 222 42 6 6 4 42
S_{4} × C_{2} 2  80 182 49  1^{2} 1_{+} 17 17 222 4 42 6 33 22 22 42

Discriminant class: −11

∞  11 17 a_{1} a_{2} a_{3}  Over p = 11  Over p = 17 2 3 5 7 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
5.  3_{i}  3_{k} 4_{+} 3 16 27 1^{2} 1_{i} 11∗ 11∗ 2^{3} 4_{+} ∗  222 33 33 222 222 222 33 222
S_{3}  24 11 31 1^{2} 1_{i} 11∗ 11∗  33 222 222 33  33 222
6.  3_{i}  5_{i} 2_{−} -3 12 -19 1^{6} 5_{i} 11∗  2^{2} 2_{−} 1 V 6 22 22 222 6 6 22 222
D_{6}  2  25 -16 5  2 0_{+} ∗ ∗ 22 22 222 6 33  222
7.  1_{k}  5_{i} 2_{+} 0 -11 -11 1^{6} 5_{i} 11∗  2 0_{+} ∗ ∗ 6 42 22 4 6 6 42 4
S_{4} × C_{2}  22 33 11  1^{2} 1_{+} 17 17 22 42 4 6 33 22 22 4

Discriminant class: 1

∞ 11 17 a_{1} a_{2} a_{3} Over p = 11  Over p = 17 2 3 5 7 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
8. 2_{−} 4_{+} 2_{−} 3 3 1 1^{3} 2_{+} ∗  2^{2} 2_{−} 1 V 33 42 22 42 33 33 42 42
S_{4}^{+} -11 -11 -11 1^{3} 2_{+} ∗ 22 42 42 33 33 22 22 42

Discriminant class: 17

∞ 11 17 a_{1} a_{2} a_{3} Over p = 11  Over p = 17 2 3 5 7 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
9. 0_{+} 4_{+} 3_{+} 3 -5 -15 2^{3} 4_{+} ∗  1^{2} 1_{+} 17 17 33 222 222 222 33 33 222 222
S_{3} -5 3 1 1^{2} 1_{+} 17 17 222 222 222 33 33 222
10. 2_{−} 2_{+} 5_{−} 1 -1 -38 2^{2} 2_{+} 1 V 1^{6} 5_{−} 17  22 6 6 222 22 22 6 222
D_{6} -77 -123 -19 2 0_{+} ∗ ∗ 222 6 222 22 33 33 222
11. 2_{−} 4_{+} 3_{−} -3 13 -21 2^{3} 4_{+} ∗  2^{2} 2_{−} 1 V 33 4 222 4 33 33 4 4
S_{4}^{−} 2  3  7 -9  1^{2} 1_{+} 17 17 222 4 4 33 33 22 22 4