Old Sextic Fields Unramified Outside of {∞, 11, 19}

Discriminant class: −19

∞  11 19 a_{1} a_{2} a_{3}  Over p = 11  Over p = 19 2 3 5 7 13 17 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}  0_{+} 5_{k} -1 2 8  1^{6} 5_{k} 19∗  6 6 33  6 33 33 6
C_{6}  -1 -5 7  222 222 6 33 33 6 6 33
2.  1_{k}  2_{−} 5_{k} -2 -11 7 1^{2} 1_{k} 11 11 1^{6} 5_{k} 19∗  6 6 33 22 6 33 33 6
A_{4} × C_{2}  22 49 11 1^{2} 1_{k} 11 11  2 2 6 33 33 6 6 33
3.  3_{i}  4_{+} 5_{k} 1 -17 -27 1^{5} 4_{+} 1  1^{6} 5_{k} 19∗  4 4 5 22 4 33 5 4
S_{5}  24  75 214 273  6 222 6 33 5 6 4 5

Discriminant class: −11

∞  11 19 a_{1} a_{2} a_{3}  Over p = 11  Over p = 19 2 3 5 7 13 17 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
4.  3_{i}  3_{k} 4_{+} 1 -4 7 1^{2} 1_{i} 11∗ 11∗ 2^{3} 4_{+} ∗  6 33 33 222 6 6 33 6
C_{6}  4  67 -122 229 1^{2} 1_{i} 11∗ 11∗  6 6 33 33 33 6
5.  1_{k}  1_{i} 4_{+} 2 -3 -9 2 0_{+} ∗  ∗ 2^{3} 4_{+} ∗  6 33 33 2 6 6 33 6
A_{4} × C_{2}  -2  3 1 1^{2} 1_{i} 11∗ 11∗  22 22 6 6 33 33 33 6

Discriminant class: 1

∞ 11 19 a_{1} a_{2} a_{3} Over p = 11 Over p = 19 2 3 5 7 13 17 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
6. 2_{−} 2_{−} 4_{+} -1 -1 -9 1^{2} 1_{k} 11 11 1^{3} 2_{+} 1  33 33 33 22 33 33 33 33
A_{4} 2 11 -7 -1 1^{2} 1_{k} 11 11 1^{3} 2_{+} 1 22 22 33 33 33 33 33 33

Discriminant class: 209

∞  11 19 a_{1} a_{2} a_{3}  Over p = 11  Over p = 19 2 3 5 7 13 17 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
7.  0_{+}  3_{k} 5_{i} -1 -55 -49 1^{2} 1_{i} 11∗ 11∗ 1^{6} 5_{i} 19  33 6 33 222 33 6 33 33
C_{6}  341 337 -107 1^{2} 1_{i} 11∗ 11∗  222 222 33 6 33 6 6 6
8.  2_{−}  1_{i} 5_{i}  1 2 11 2 0_{+} ∗  ∗ 1^{6} 5_{i} 19  33 6 33 2 33 6 33 33
A_{4} × C_{2}  18 24  7 1^{2} 1_{i} 11∗ 11∗  2 2 33 6 33 6 6 6