Quartic Fields Ramifying Above {5, 23}

Discriminant class: −23

∞ 5 23 a_{1} a_{2} a_{3}  Over p = 5  Over p = 23  2 3 7 11 13 17 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 9 10 1^{2} 1_{−} 5∗ 5∗ 1^{4} 3_{k} 23∗ 23 3 3 2 4 3 4 2 3
S_{4} 2 -15 1^{2} 1_{+} 5 5 3 4 3 4 3 4 22 2

Discriminant class: 1

∞ 5 23 a_{1} a_{2} a_{3} Over p = 5  Over p = 23 2 3 7 11 13 17 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 9 0 2^{2} 2_{−} 1 V 2^{2} 2_{+} 1 V 22 22 22 22 22 22 22
V_{4} 3 49 22 22 22 22 22

Discriminant class: 5

∞ 5 23 a_{1} a_{2} a_{3} Over p = 5  Over p = 23 2 3 7 11 13 17 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. 0_{+} 3_{−} 2_{−} 1 -29 -29 1^{4} 3_{−} 5 5 2^{2} 2_{−} ∗ ∗ 4 4 4 22 4 4  22
C_{4}  151  4  4 4 4 22 22