Quartic Fields Unramified Outside of {∞, 19, 31}

Discriminant class: −31

∞ 19 31 a_{1} a_{2} a_{3}  Over p = 19  Over p = 31  2 3 5 7 11 13 17 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} 2 -45 -46 1^{2} 1_{i} 19∗ 19∗ 1^{4} 3_{k} 31∗ 31 22 2  22 4 2 4 4
D_{4} -60 1^{2} 1_{i} 19∗ 19∗ 2 2 22 4 22 2 22 4

Discriminant class: 1

∞ 19 31 a_{1} a_{2} a_{3}  Over p = 19  Over p = 31 2 3 5 7 11 13 17 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. 2_{−} 2_{−} 2_{+} 0 25 0 1^{2} 1_{i} 19∗ 19∗ 2^{2} 2_{+} 1 V 22 22  22 22 22 22
V_{4} 3 9 1^{2} 1_{i} 19∗ 19∗ 22 22 22 22 22 22 22
3. 0_{+} 2_{+} 2_{+} 0 -17 -31 1^{3} 2_{+} 1  2^{2} 2_{+} 1 V 3 3 3 22 22 3 3 3
A_{4} -13 3 22 3 3 3 3 3 3

Discriminant class: 589

∞ 19 31 a_{1} a_{2} a_{3}  Over p = 19  Over p = 31  2 3 5 7 11 13 17 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_{−} 1_{i} 3_{i}  1 -19 -29 1^{2} 1_{i} 19∗ 19∗ 1^{4} 3_{i} 31 31∗ 2 22  22 4 22 4 4
D_{4} 3  190  22 22 2 4 22 22 2 4