Quartic Fields Ramifying Above {23, 37}

Discriminant class: −23

∞ 23 37 a_{1} a_{2} a_{3}  Over p = 23  Over p = 37  2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 29 31 41 43 47 53 59 61
1. 1_{k} 3_{k} 2_{−}  1 -37 -125 1^{4} 3_{k} 23∗ 23 1^{2} 1_{+} 37 37 3 3 4 2 2 3 2 2
S_{4} 2 -544 1^{2} 1_{−} 37∗ 37∗ 3 3 3 2 3 2 22 2

Discriminant class: 1

∞ 23 37 a_{1} a_{2} a_{3}  Over p = 23  Over p = 37 2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 41 43 47 53 59 61
2. 2_{−} 2_{+} 2_{−} 0 -7 0 2^{2} 2_{+} 1 V 2^{2} 2_{−} 1 V 22  22 22 22 22 22 22
V_{4} 15 225 22 22 22 22 22 22
3. 2_{−} 2_{−} 2_{+} 2 7 -17 1^{2} 1_{k} 23 23 1^{3} 2_{+} 1  3 3 3 3 22 3 3 3
A_{4} 4 9 1^{2} 1_{k} 23 23 22 22 3 22 22 3 3 3

Discriminant class: 37

∞ 23 37  a_{1} a_{2} a_{3} Over p = 23  Over p = 37 2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 41 43 47 53 59 61
4. 0_{+} 2_{−} 3_{−}  -1 -217 215 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{−} 37 37 4 22 4 22  4 4 4
C_{4}  10483  4 4 22 4  22 4 4