Quartic Fields Ramifying Above {17, 31}

Discriminant class: −527

∞ 17 31 a_{1} a_{2} a_{3}  Over p = 17  Over p = 31 2 3 5 7 11 13 19 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 37 41 43 47 53 59 61
1. 1_{k} 3_{−} 1_{k} 2 -7 -25 1^{4} 3_{−} 17∗ 17∗ 1^{2} 1_{k} 31 31 3 3 4 2 3 4 22 3
S_{4} 4 -18 22 3 4 4 3 2 3 3

Discriminant class: −31

∞ 17 31 a_{1} a_{2} a_{3}  Over p = 17  Over p = 31  2 3 5 7 11 13 19 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 37 41 43 47 53 59 61
2. 1_{k} 2_{−} 3_{k}  -1 12 60 1^{2} 1_{−} 17∗ 17∗ 1^{4} 3_{k} 31∗ 31 3 4 3 3 2 4 3 4
S_{4} 2 -616 1^{2} 1_{+} 17 17 2 4 3 2 22 4 3 4

Discriminant class: 1

∞ 17 31 a_{1} a_{2} a_{3} Over p = 17  Over p = 31 2 3 5 7 11 13 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 29 37 41 43 47 53 59 61
3. 2_{−} 2_{−} 2_{+} 0 7 0 2^{2} 2_{−} 1 V 2^{2} 2_{+} 1 V  22 22 22 22 22  22
V_{4} 27  144  22 22 22 22  22  22

Discriminant class: 17

∞ 17 31 a_{1} a_{2} a_{3}  Over p = 17  Over p = 31 2 3 5 7 11 13 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 29 37 41 43 47 53 59 61
4. 2_{−} 3_{+} 2_{−}  -1 130 1 1^{4} 3_{+} 17 17 2^{2} 2_{−} ∗ ∗ 22 4 4 4 4 22 22 4
C_{4} 10  4217  4 4 4  22 4