Quartic Fields Ramifying Above {37, 43}

Discriminant class: 1

∞ 37 43 a_{1} a_{2} a_{3} Over p = 37  Over p = 43  2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 23 29 31 41 47 53 59 61
1. 2_{−} 2_{−} 2_{+} 0 3 0 2^{2} 2_{−} 1 V 2^{2} 2_{+} 1  V 22 22 22 22  22 22 22
V_{4} 11  400  22 22 22  22 22
2. 2_{−} 2_{+} 2_{−} 2 3 -41 1^{3} 2_{+} 1  1^{2} 1_{i} 43∗ 43∗ 3 3 3 3 22 3 3 3
A_{4} 4 173 1^{2} 1_{i} 43∗ 43∗ 22 22 3 3 3 3

Discriminant class: 37

∞ 37 43  a_{1} a_{2} a_{3}  Over p = 37  Over p = 43 2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 23 29 31 41 47 53 59 61
3. 0_{+} 3_{−} 2_{−}  1 -402 -400 1^{4} 3_{−} 37 37 2^{2} 2_{−} ∗ ∗ 4  4 22 22 4 4 4
C_{4}  37493  4 4 4 22  4 4