Old Sextic Fields Ramifying Above {5, 13}

Discriminant class: 1

∞ 5 13 a_{1} a_{2} a_{3}  Over p = 5  Over p = 13  2 3 7 11 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. 2_{−} 2_{−} 4_{+} 1 -1 3 2 0_{+} ∗ ∗ 1^{3} 2_{+} 1  33 33 33 33 33 33 33 33
A_{4}  -1 1 1 1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} 1  22 33 33 33 22 22 33 33
2. 2_{−} 6_{+} 2_{−} -1 -5 -5 1^{5} 6_{+} 1  2 0_{+} ∗  ∗ 5 33 33 5 33 33 33 5
A_{5}  0 3 2  1^{2} 1_{−} 13∗ 13∗ 5 22 33 5 5 5 33 22

Discriminant class: 5

∞  5 13 a_{1} a_{2} a_{3}  Over p = 5  Over p = 13 2 3 7 11 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
3.  0_{+}  3_{+} 4_{+} -1 -12 13 1^{2} 1_{+} 5 5 2^{3} 4_{+} ∗  6 6 6 33 6 33 6 33
C_{6}  19 -10  -5 1^{2} 1_{+} 5 5  6 33 6 222 222 33 33
4.  2_{−}  1_{−} 4_{+} -2  0  1 1^{2} 1_{−} 5∗ 5∗ 2^{3} 4_{+} ∗  6 6 6 33 6 33 6 33
A_{4} × C_{2}  0 -2  1 2 0_{+} ∗ ∗  22 6 33 6 2 2 33 33
5.  2_{−}  7_{−} 2_{+} -1  0  0 1^{5} 7_{−} 5  2^{2} 2_{+} ∗ ∗ 6 6 222 5 222 5 4 5
S_{5}  0 -4  -1  33 6 5 4 4 6 33 33
6.  2_{−}  7_{−} 4_{+} -1 15  -5 1^{5} 7_{−} 5  2^{3} 4_{+} ∗  4 4 222 5 6 22 4 33
S_{5}  3  -35 22  8  33 6 22 6 6 4 5 22
7.  2_{−}  7_{−} 4_{+}  3  5 -20 1^{5} 7_{−} 5  1^{5} 4_{+} ∗  222 4 6 33 6 5 6 33
S_{5}  -100 -160 -80  33 4 33 6 222 4 5 33
8.  2_{−}  9_{−} 4_{+} -2 -5 20 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  6 6 6 5 4 5 4 33
S_{5}  -25 -51 127  22 4 33 222 6 6 5 5
9.  2_{−}  9_{−} 4_{+} -1  0 -125 1^{5} 9_{−} 5  2^{3} 4_{+} ∗  4 222 4 5 6 33 4 33
S_{5}  3  0 875 1000  22 4 5 4 6 4 5 33

Discriminant class: 13

∞  5 13 a_{1} a_{2} a_{3}  Over p = 5  Over p = 13 2 3 7 11 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.  2_{−}  2_{−} 5_{+} 2 6 -6 1^{2} 1_{−} 5∗ 5∗ 1^{6} 5_{+} 13  6 33 6 6 33 6 33 33
A_{4} × C_{2} 2  -8 -21 1 1^{2} 1_{+} 5 5  2 6 6 33 2 22 6 33
11.  2_{−}  6_{+} 3_{−} -1 5 10 1^{5} 6_{+} ∗  1^{4} 3_{−} 13 13 6 5 4 4 33 222 33 22
S_{5}  -5 26 -11  222 222 6 5 6 22 4 33

Discriminant class: 65

∞  5 13 a_{1} a_{2} a_{3}  Over p = 5  Over p = 13 2 3 7 11 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
12.  0_{+}  3_{−} 5_{−} -1 -18 17 1^{2} 1_{−} 5∗ 5∗ 1^{6} 5_{−} 13∗  33 6 33 6 6 6 6 33
C_{6}  2  58 -16 -1 1^{2} 1_{−} 5∗ 5∗  222 33 6 6  222 6 33
13.  2_{−}  1_{+} 5_{−} -3  7 -9 1^{2} 1_{+} 5 5 1^{6} 5_{−} 13∗  33 6 33 6 6 6 6 33
A_{4} × C_{2} 2  5 -1 -1 2 0_{+} ∗ ∗  2 33 6 6 22 2 6 33

Discriminant class: 65 (continued)

∞  5 13 a_{1} a_{2} a_{3}  Over p = 5  Over p = 13 2 3 7 11 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
14.  2_{−}  5_{+} 5_{−} -1 -5 30 1^{5} 5_{+} 5∗  1^{6} 5_{−} 13∗  5 6 22 222 4 6 4 33
S_{5}  6  -85 127 -92  4 22 4 6 33 6 4 33
15.  2_{−}  7_{+} 5_{−}  0  0 -65 1^{5} 7_{+} 5∗  1^{6} 5_{−} 13∗  33 4 5 4 6 4 4 5
S_{5}  6  130 169 -195  222 5 4 6 33 4 4 22
16.  2_{−}  7_{+} 5_{−}  0  0 -65 1^{5} 7_{+} 5∗  1^{6} 5_{−} 13∗  5 4 5 4 6 4 4 5
S_{5}  6  -195 -416 -195  4 5 4 222 5 6 4 5