Low Degree Fields Ramifying Above {7, 13, 17}

Discriminant class: −1547

∞ 7 13 17 a_{1} a_{2} a_{3}  Over p = 7  Over p = 13  Over p = 17  2 3 5 11 19 23 29 31
G h  f^{e} c_{w} d s f^{e} c_{w} d  s f^{e} c_{w} d  s 37 41 43 47 53 59 61 67
1. 1_{k} 1_{i} 1_{−} 1_{−} 0 1547  1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{−} 17∗ 17∗ 2  2  2 2 2
C_{2} 12  2  2
2. 1_{k} 1_{i} 1_{−} 1_{−} -1  -1 8 1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{−} 17∗ 17∗ 2 3 2 3 3 2 2 2
S_{3} 3 2 3 3 3 2

Discriminant class: −91

∞ 7 13 17 a_{1} a_{2} a_{3} Over p = 7  Over p = 13  Over p = 17 2 3 5 11 19 23 29 31
G h  f^{e} c_{w} d s f^{e} c_{w} d  s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
3. 1_{k} 1_{k} 1_{−} 2_{+} 0 -17 -68 1^{2} 1_{k} 7 7 1^{2} 1_{−} 13∗ 13∗ 1^{3} 2_{+} ∗  2 2 3 2 3  3 3
S_{3}  2 3 3  3 2 2