Quartic Fields Ramifying Above {31, 47}

Discriminant class: −31

∞ 31 47 a_{1} a_{2} a_{3}  Over p = 31  Over p = 47  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 43 53 59 61
1. 1_{k} 3_{k} 2_{−}  0 31  0 1^{4} 3_{k} 31∗ 31 1^{2} 1_{i} 47∗ 47∗ 22 4 22 22 2 2 4 22
D_{4} 3  -124  1^{2} 1_{i} 47∗ 47∗ 2 2 4 22 2 4  4
2. 1_{k} 1_{i} 2_{+} -1 -2  7 1^{2} 1_{i} 31∗ 31∗ 2^{2} 2_{+} 1  V 3 4 3 3 4 4 4 3
S_{4}  2  2 2 2 3 4 4 3 2
3. 1_{k} 3_{k} 2_{−} -1 -50 -157 1^{4} 3_{k} 31∗ 31 1^{2} 1_{k} 47 47 3 2 3 3 2 4 4 3
S_{4} 2 -120 1^{2} 1_{k} 47 47 4 4 4 3 2 2 3 2
4. 1_{k} 3_{k} 2_{−} -1 12 -188 1^{4} 3_{k} 31∗ 31 1^{2} 1_{i} 47∗ 47∗ 3 4 3 3 4 2 2 3
S_{4} 2 376 1^{2} 1_{i} 47∗ 47∗ 4 4 4 3 4 4 3 2

Discriminant class: 1

∞ 31 47 a_{1} a_{2} a_{3} Over p = 31  Over p = 47  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 43 53 59 61
5. 2_{−} 2_{+} 2_{−} 0 39 0 2^{2} 2_{+} 1 V 1^{2} 1_{i} 47∗ 47∗  22 22  22 22 22 22
V_{4} 15  16  1^{2} 1_{i} 47∗ 47∗ 22 22 22 22 22 22  22
6. 2_{−} 2_{+} 2_{−} -1 -30 21 1^{3} 2_{+} 1  1^{2} 1_{i} 47∗ 47∗ 22 3 3 3 3 3 3 3
A_{4} 4 253 1^{2} 1_{i} 47∗ 47∗ 22 3 3 3 3 3

Discriminant class: 1457

∞ 31 47 a_{1} a_{2} a_{3}  Over p = 31  Over p = 47  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 43 53 59 61
7. 2_{−} 3_{i} 1_{i} -2 17 -16 1^{4} 3_{i} 31 31∗ 1^{2} 1_{i} 47∗ 47∗ 22 4 2 22 22 22 4 2
D_{4} 9  95  22 22 4 2 22 4  4