Low Degree Fields Ramifying Above {19, 53}

Discriminant class: −1007

∞ 19 53 a_{1} a_{2} a_{3}  Over p = 19  Over p = 53  2 3 5 7 11 13 17 23
G h  f^{e} c_{w} d s f^{e} c_{w} d  s 29 31 37 41 43 47 59 61
1. 1_{k} 1_{k} 1_{−} 0 1007  1^{2} 1_{k} 19 19 1^{2} 1_{−} 53∗ 53∗  2  2  2
C_{2} 30  2  2  2 2
2. 1_{k} 1_{k} 1_{−} -1  -2 7 1^{2} 1_{k} 19 19 1^{2} 1_{−} 53∗ 53∗ 3 3 2  3 2 3 2
S_{3}  2 3 2  3  2 2

Discriminant class: −19

∞ 19 53 a_{1} a_{2} a_{3}  Over p = 19  Over p = 53 2 3 5 7 11 13 17 23
G h  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 37 41 43 47 59 61
3. 1_{k} 1_{i} 2_{+} -1 -35 104 1^{2} 1_{i} 19∗ 19∗ 1^{3} 2_{+} ∗  2 2 3 3 3 2
S_{3}  2 2 2 2  2 3