Low Degree Fields Ramifying Above {31, 53}

Discriminant class: −1643

∞ 31 53 a_{1} a_{2} a_{3}  Over p = 31  Over p = 53  2 3 5 7 11 13 17 19
G h  f^{e} c_{w} d s f^{e} c_{w} d  s 23 29 37 41 43 47 59 61
1. 1_{k} 1_{k} 1_{−} 0 1643  1^{2} 1_{k} 31 31 1^{2} 1_{−} 53∗ 53∗ 2  2  2 2 2 2
C_{2} 10  2 2 2 2

Discriminant class: −31

∞ 31 53 a_{1} a_{2} a_{3}  Over p = 31  Over p = 53 2 3 5 7 11 13 17 19
G h  f^{e} c_{w} d  s f^{e} c_{w} d s 23 29 37 41 43 47 59 61
2. 1_{k} 1_{i} 2_{+} 0 -53 -159 1^{2} 1_{i} 31∗ 31∗ 1^{3} 2_{+} ∗  3 2 3 3 2 2 2
S_{3} 3  2 2 2 3 2 3 3 2
3. 1_{k} 1_{i} 2_{+} -1 18 104 1^{2} 1_{i} 31∗ 31∗ 1^{3} 2_{+} ∗ 2 3 2 2 2 3
S_{3} 3  2 2 2 3 2 3 3 2
4. 1_{k} 1_{i} 2_{+} -1 18 -55 1^{2} 1_{i} 31∗ 31∗ 1^{3} 2_{+} ∗  3 2  3 2 2 2 3
S_{3} 3  2 2 2  2 3  2