Quartic Fields Unramified Outside of {∞, 3, 31}

Discriminant class: −31

∞ 3 31 a_{1} a_{2} a_{3} Over p = 3  Over p = 31  2 5 7 11 13 17 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} 2_{−} 3_{k} -1 12 29 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{k} 31∗ 31 3 3 3 2 2 4 3 2
S_{4} 2 4 4 2 3 4 22 2 3 4

Discriminant class: −3

∞ 3 31 a_{1} a_{2} a_{3}  Over p = 3  Over p = 31 2 5 7 11 13 17 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} 3_{i} 2_{+} 0 9 0 1^{4} 3_{i} 3∗ 3 2^{2} 2_{+} 1 V 4 4  2 22 2 22 2
D_{4} -3 2 22 4 22 4 2 4 22

Discriminant class: 1

∞ 3 31 a_{1} a_{2} a_{3} Over p = 3  Over p = 31  2 5 7 11 13 17 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_{−} -1 -7 -8 2^{2} 2_{+} 1 V 1^{2} 1_{i} 31∗ 31∗ 22 22  22 22 22  22
V_{4} 3  64  1^{2} 1_{i} 31∗ 31∗ 22 22 22 22 22 22 22 22

Discriminant class: 93

∞ 3 31 a_{1} a_{2} a_{3}  Over p = 3  Over p = 31  2 5 7 11 13 17 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_{k} 1_{k} 2 6 5 1^{4} 3_{k} 3 3∗ 1^{2} 1_{k} 31 31 4 4  22 2 22 22 22
D_{4} 7 2 0_{+} ∗ ∗ 22 2 4 2 4 22 4 2
5. 2_{−} 3_{k} 3_{k} -1 12 -2 1^{3} 3_{k} 3  1^{4} 3_{k} 31∗ 31 4 2 3 3 4 3 3 3
S_{4} 2  35  3 2 4 2 2 22 4 4