Old Sextic Fields Ramifying Above {7, 13}

Discriminant class: −91

∞  7 13 a_{1} a_{2} a_{3} Over p = 7  Over p = 13  2 3 5 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.  3_{i}  5_{i} 3_{−} -1 22 -22 1^{6} 5_{i} 7  1^{2} 1_{−} 13∗ 13∗ 6 6 33 6 6 33 33
C_{6}  8  148 -148 337  1^{2} 1_{−} 13∗ 13∗ 33 6  33 33 33 6
2.  3_{i}  3_{i} 5_{−} 1 21 22 3^{2} 3_{i} 7  1^{6} 5_{−} 13∗  6 6  6 6 33 33 33
C_{6}  14  58 -23 155  6 33 33  33 6
3.  3_{i}  5_{i} 5_{−} 1  8 22 1^{6} 5_{i} 7  1^{6} 5_{−} 13∗  6 222 33 222 6  33 33
C_{6}  6  -20 -426 1611  33 6 33 33 33 33 33 222
4.  3_{i}  5_{i} 5_{−} -1  8 -113 1^{6} 5_{i} 7  1^{6} 5_{−} 13∗  222 6 33 6 222 33  33
C_{6}  6  435 -666 428  33 222 33 33 33 33  6
5.  1_{k}  5_{i} 1_{+} 3  2  -1 1^{6} 5_{i} 7  1^{2} 1_{+} 13 13 6 6 33 6 6 33 33 22
A_{4} × C_{2}  -3 -2  1  2 0_{+} ∗  ∗ 33 6 22 22 33 33 33 6

Discriminant class: −7

∞  7 13 a_{1} a_{2} a_{3}  Over p = 7  Over p = 13  2 3 5 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
6.  3_{i}  3_{k} 4_{+} 1 -3 -7 3^{2} 3_{k} 7∗  2^{3} 4_{+} ∗  33 6 222 33 6 6 33 33
C_{6}  25 55 79  222 33 6 33 222  6 6
7.  3_{i}  5_{k} 4_{+} 1  1 -13 1^{6} 5_{k} 7∗  2^{3} 4_{+} ∗  6 6 33 222 6  33
C_{6}  39  162 400 736  6  6 33 6 33 222 6
8.  3_{i}  5_{k} 4_{+} 1  1 -13 1^{6} 5_{k} 7∗  2^{3} 4_{+} ∗  33 222 6  6 222 33 33
C_{6}  3  71 -419 827  6 33 6 33 6 33 6 222
9.  1_{k}  5_{k} 2_{−} -2 -3 -1 1^{6} 5_{k} 7∗  1^{2} 1_{+} 13 13 33 6 6 33 6 6 33 22
A_{4} × C_{2}  2 24 8  1^{2} 1_{−} 13∗ 13∗ 6 33 2 22 6 33 6 6

Discriminant class: 1

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

Discriminant class: 13

∞  7 13 a_{1} a_{2} a_{3} Over p = 7  Over p = 13  2 3 5 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
11.  0_{+}  4_{+} 3_{+} 1 -14 -9 2^{3} 4_{+} ∗  1^{2} 1_{+} 13 13 6 33 6 6 33 6 33
C_{6}  35 16 -1  1^{2} 1_{+} 13 13 6 6 222  6 33 6 33
12.  0_{+}  4_{+} 5_{+} -1 -31  4 2^{3} 4_{+} ∗  1^{6} 5_{+} 13  6  6 222 33 222 33 33
C_{6}  3  162 -81 -27  6 6 6 33 6 33 6
13.  0_{+}  4_{+} 5_{+} -1 -31  4 2^{3} 4_{+} ∗  1^{6} 5_{+} 13  222 33 6 6  6  33
C_{6}  3  253 101 -391  6 222 6 33 6 33 222 33

Discriminant class: 13 (continued)

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