Quartic Fields Unramified Outside of {∞, 31, 43}

Discriminant class: −43

∞ 31 43 a_{1} a_{2} a_{3} Over p = 31  Over p = 43  2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 23 29 37 41 47 53 59 61
1. 1_{k} 2_{+} 3_{i}  2 -20 -21 2^{2} 2_{+} 1 V 1^{4} 3_{i} 43∗ 43 4 2 4 4 22 22 22 4
D_{4} -223 22 2 2 22 22 22 2

Discriminant class: −31

∞ 31 43 a_{1} a_{2} a_{3}  Over p = 31  Over p = 43 2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 23 29 37 41 47 53 59 61
2. 1_{k} 3_{k} 2_{−}  1 43 188 1^{4} 3_{k} 31∗ 31 2^{2} 2_{−} ∗ ∗ 3 2 3 3 2 4 4 3
S_{4} 2 -647 2 4 4 3 22 2 3 4

Discriminant class: 1

∞ 31 43 a_{1} a_{2} a_{3}  Over p = 31  Over p = 43 2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 23 29 37 41 47 53 59 61
3. 2_{−} 2_{−} 2_{+} 0 37 0 1^{2} 1_{i} 31∗ 31∗ 2^{2} 2_{+} 1 V 22 22 22 22 22 22 22 22
V_{4} 3  9  1^{2} 1_{i} 31∗ 31∗  22 22 22  22  22
4. 0_{+} 2_{+} 2_{+} -1 -22 -1 1^{3} 2_{+} 1  1^{3} 2_{+} 1  22 3 22 22 3 3 22 3
A_{4} 2  71  3 3 3 3 22 22 3 3

Discriminant class: 1333

∞ 31 43 a_{1} a_{2} a_{3}  Over p = 31  Over p = 43  2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 23 29 37 41 47 53 59 61
5. 2_{−} 1_{k} 3_{k}  1 -5 -51 1^{2} 1_{k} 31 31 1^{4} 3_{k} 43 43∗ 4 22 4 4 2 2 2 4
D_{4}  107  2 0_{+} ∗ ∗  2 22 22 22  2 22 22