Quartic Fields Unramified Outside of {∞, 23, 31}

Discriminant class: −31

∞ 23 31 a_{1} a_{2} a_{3} Over p = 23  Over p = 31  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 37 41 43 47 53 59 61
1. 1_{k} 2_{−} 3_{k}  -1 12 -95 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{k} 31∗ 31 3 2 3 3 4 4 4 3
S_{4} 2 -244 4 4 3 4 22 4 3 2

Discriminant class: −23

∞ 23 31 a_{1} a_{2} a_{3}  Over p = 23  Over p = 31  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 37 41 43 47 53 59 61
2. 1_{k} 3_{k} 2_{−}  2 -33 -34 1^{4} 3_{k} 23∗ 23 1^{2} 1_{i} 31∗ 31∗  22 4 4 2 22 2 4
D_{4}  -424  1^{2} 1_{i} 31∗ 31∗ 22 2 22 2 22 2 22 2

Discriminant class: 1

∞ 23 31 a_{1} a_{2} a_{3}  Over p = 23  Over p = 31  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 37 41 43 47 53 59 61
3. 2_{−} 2_{+} 2_{−} 0 27 0 2^{2} 2_{+} 1 V 1^{2} 1_{i} 31∗ 31∗  22 22 22 22 22 22 22
V_{4} 9 4 1^{2} 1_{i} 31∗ 31∗ 22 22 22 22 22
4. 2_{−} 2_{−} 2_{+} -1 20 -7 1^{2} 1_{k} 23 23 1^{3} 2_{+} 1  22 3 3 3 3 3 3 3
A_{4} 4 95 1^{2} 1_{k} 23 23 22 3 3 3 22 3 3

Discriminant class: 713

∞ 23 31 a_{1} a_{2} a_{3}  Over p = 23  Over p = 31  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 37 41 43 47 53 59 61
5. 2_{−} 3_{i} 1_{i}  1 -14 -56 1^{4} 3_{i} 23 23∗ 1^{2} 1_{i} 31∗ 31∗  2 4 4 22 2 22 4
D_{4} 3  192  2 22 22 22 22 22 22 22