Quartic Fields Ramifying Above {23, 29}

Discriminant class: −667

∞ 23 29 a_{1} a_{2} a_{3}  Over p = 23  Over p = 29 2 3 5 7 11 13 17 19
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} 1_{k} 3_{−} 1 4 -38 1^{2} 1_{k} 23 23 1^{4} 3_{−} 29 29 4 4 2 2 22  22 22
D_{4} -35 2 0_{+} ∗ ∗ 4 22 4 22 4 2 22 22

Discriminant class: −23

∞ 23 29 a_{1} a_{2} a_{3}  Over p = 23  Over p = 29 2 3 5 7 11 13 17 19
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. 1_{k} 1_{k} 2_{−} 1 0 -11 1^{2} 1_{k} 23 23 2^{2} 2_{−} 1 V 22 22 2 2 4 22 4 4
D_{4} 5 2 0_{+} ∗ ∗ 22 4 22 4 22 2 22 4

Discriminant class: 1

∞ 23 29 a_{1} a_{2} a_{3}  Over p = 23  Over p = 29 2 3 5 7 11 13 17 19
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_{−} 2_{−} 2_{+} 0 -3 0 1^{2} 1_{i} 23∗ 23∗ 1^{2} 1_{+} 29 29 22 22 22 22 22  22 22
V_{4} 6 169 1^{2} 1_{i} 23∗ 23∗ 1^{2} 1_{+} 29 29 22 22 22 22 22 22 22

Discriminant class: 29

∞ 23 29 a_{1} a_{2} a_{3}  Over p = 23  Over p = 29  2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
4. 0_{+} 2_{−} 3_{−} 1 -170 -502 1^{2} 1_{i} 23∗ 23∗ 1^{4} 3_{−} 29 29 4 4  22 4 22 4 4
C_{4} 545 1^{2} 1_{i} 23∗ 23∗ 4 4 4 4 4 22 4
5. 2_{−} 2_{+} 1_{−} 1 -2 -4 2^{2} 2_{+} 1  V 1^{2} 1_{−} 29∗ 29∗ 2 2 22 22 4 22 4 4
D_{4} 3  16  2 0_{+} ∗  ∗ 2 4 2 4 2 22 22 4
6. 2_{−} 2_{+} 3_{−} 2 16 15 2^{2} 2_{+} 1  V 1^{4} 3_{−} 29 29 4 4 22 22 2  2 2
D_{4} 4  223  4 2 4 2 4 22 22 2