Quartic Fields Ramifying Above {17, 23}

Discriminant class: −23

∞ 17 23 a_{1} a_{2} a_{3}  Over p = 17  Over p = 23  2 3 5 7 11 13 19 29
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 1_{k} 2_{−} 3_{k} -1 -14 56 1^{2} 1_{+} 17 17 1^{4} 3_{k} 23∗ 23 3 3 2 4 4 3 4 3
S_{4} 2 8 1^{2} 1_{−} 17∗ 17∗ 3 4 3 4 3 4 22 4

Discriminant class: 1

∞ 17 23 a_{1} a_{2} a_{3} Over p = 17  Over p = 23 2 3 5 7 11 13 19 29
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
2. 2_{−} 2_{−} 2_{+} 0 3 0 2^{2} 2_{−} 1 V 2^{2} 2_{+} 1 V  22 22 22 22  22 22
V_{4} 21  100  22 22 22 22  22  22

Discriminant class: 17

∞ 17 23 a_{1} a_{2} a_{3}  Over p = 17  Over p = 23 2 3 5 7 11 13 19 29
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
3. 2_{−} 3_{+} 2_{−}  1 96 -1 1^{4} 3_{+} 17 17 2^{2} 2_{−} ∗ ∗ 22 4 4 4 4  4
C_{4} 10  2347  4 4 4  22 4