Quartic Fields Unramified Outside of {∞, 5, 31}

Discriminant class: −155

∞ 5 31 a_{1} a_{2} a_{3} Over p = 5  Over p = 31 2 3 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} 3_{−} 1_{k} 2 4 3 1^{4} 3_{−} 5 5 1^{2} 1_{k} 31 31 4 22 4 2 22 22  22
D_{4} -9 2 0_{+} ∗ ∗ 2 22 22 4 22 22 2

Discriminant class: −31

∞ 5 31 a_{1} a_{2} a_{3} Over p = 5  Over p = 31 2 3 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} 2_{−} 1_{k} 1 0 3 2^{2} 2_{−} 1 V 1^{2} 1_{k} 31 31 22 4 22 2 4 4 22 4
D_{4} -1 2 0_{+} ∗ ∗ 2 4 4 22 4 2

Discriminant class: 1

∞ 5 31 a_{1} a_{2} a_{3} Over p = 5  Over p = 31  2 3 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_{−} 0 13 0 1^{2} 1_{+} 5 5 1^{2} 1_{i} 31∗ 31∗ 22 22 22 22 22 22  22
V_{4} 6 81 1^{2} 1_{+} 5 5 1^{2} 1_{i} 31∗ 31∗ 22 22 22 22 22 22

Discriminant class: 5

∞ 5 31 a_{1} a_{2} a_{3}  Over p = 5  Over p = 31  2 3 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_{−} 0_{+} -1 1 -1 1^{4} 3_{−} 5 5  4 4 4  4 4 22 4
C_{4}  1  22 4  4 4 4 22
5. 0_{+} 3_{−} 2_{−} 1 -39 -39 1^{4} 3_{−} 5 5 1^{2} 1_{i} 31∗ 31∗ 4 4 4 22 4 4 22 4
C_{4} 281 1^{2} 1_{i} 31∗ 31∗ 4 4 4 4 22 22
6. 2_{−} 1_{−} 2_{+} 2 2 1 1^{2} 1_{−} 5∗ 5∗ 2^{2} 2_{+} 1  V 2 4 2 22 4 4 22 4
D_{4} 3  8  2 0_{+} ∗ ∗  22 4  4 2 4  22
7. 2_{−} 3_{−} 2_{+} 0 5 0 1^{4} 3_{−} 5 5 2^{2} 2_{+} 1  V 4 2 4 22 2 2  2
D_{4} 4  45  22 2  2 4 2 22 22