Old Sextic Fields Ramifying Above {11, 13}

Discriminant class: −143

∞  11 13 a_{1} a_{2} a_{3} Over p = 11  Over p = 13  2 3 5 7 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}  3_{i} 5_{−} -1 34 -35 3^{2} 3_{i} 11  1^{6} 5_{−} 13∗  33 33 222 33 6 33 33 6
C_{6}  10  162 36 467  222 6 33 6 222  6 6
2.  3_{i}  5_{k} 3_{+} -3 23 -41 1^{6} 5_{k} 11  2^{2} 2_{−} 1  V 22 33 222 22 6 22 33 6
D_{6}  10  80 -60 16  1^{2} 1_{−} 13∗ 13∗ 222 222 22 6 222 33 222 6
3.  1_{k}  5_{k} 3_{−} 3 1 -3 1^{6} 5_{k} 11  1^{4} 3_{−} 13 13 42 33 222 22 6 42 33 6
S_{4} × C_{2}  -30 -28 49  2 0_{+} ∗  ∗ 222 4 22 6 4 33 222 6

Discriminant class: −11

∞  11 13 a_{1} a_{2} a_{3}  Over p = 11  Over p = 13  2 3 5 7 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
4.  3_{i}  3_{k} 4_{+} -1 0  5 3^{2} 3_{k} 11∗  2^{3} 4_{+} ∗  6 33  6 6 6 33 6
C_{6}  7  39 -74 155  33 6 6  33 6
5.  3_{i}  5_{i} 2_{−}  1 5  6 1^{6} 5_{i} 11∗  2^{2} 2_{−} 1  V 222 33 22 222 6 222 33 6
D_{6}  20 16 64  2 0_{+} ∗  ∗ 22 22 222 6 22 33 22 6
6.  1_{k}  5_{i} 4_{+}  2 -2 -40 1^{6} 5_{i} 11∗  1^{4} 3_{−} 13 13 4 33 22 222 6 4 33 6
S_{4} × C_{2} 2  -142 413 -194  1^{2} 1_{−} 13∗ 13∗ 22 42 222 6 42 33 22 6

Discriminant class: 1

∞ 11 13 a_{1} a_{2} a_{3} Over p = 11  Over p = 13 2 3 5 7 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
7. 2_{−} 4_{+} 4_{−} -1 0 12 1^{3} 2_{+} ∗  1^{4} 3_{−} 13 13 42 33 22 22 33 42 33 33
S_{4}^{+} 4 -11 30 -4 1^{3} 2_{+} ∗ 1^{2} 1_{+} 13 13 22 42 22 33 42 33 22 33

Discriminant class: 13

∞  11 13 a_{1} a_{2} a_{3} Over p = 11  Over p = 13 2 3 5 7 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
8.  0_{+}  4_{+} 3_{+} -3 -12 29 2^{3} 4_{+} ∗  1^{2} 1_{+} 13 13 222 33 222 222 33 222 33 33
S_{3}  5 -20 -4  1^{2} 1_{+} 13 13 222 222 222 33 222 33 222 33
9.  0_{+}  4_{+} 5_{+} -2 -46 84 2^{3} 4_{+} ∗  1^{6} 5_{+} 13  6 3 222 6 3 6 3 3
S_{3} × C_{3} 3  122 -83 -51  222 6 6 3 222 33 6 3
10.  2_{−}  4_{+} 3_{−} 3 4 3 2^{3} 4_{+} ∗  1^{4} 3_{−} 13 13 4 33 222 222 33 4 33 33
S_{4}^{−}  -6 -7 -2  222 4 222 33 4 33 222 33