Quartic Fields Ramifying Above {43, 47}

Discriminant class: −43

∞ 43 47 a_{1} a_{2} a_{3}  Over p = 43  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 31 37 41 53 59 61
1. 1_{k} 3_{i} 2_{+} 0 -43 0 1^{4} 3_{i} 43∗ 43 2^{2} 2_{+} 1 V 4 4 2 4 22 22  2
D_{4} 3 -43 22 2 22 4 22 4

Discriminant class: 1

∞ 43 47 a_{1} a_{2} a_{3} Over p = 43  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 31 37 41 53 59 61
2. 2_{−} 2_{+} 2_{−} 0 45 0 2^{2} 2_{+} 1 V 1^{2} 1_{i} 47∗ 47∗ 22 22 22 22 22 22  22
V_{4} 15 1 1^{2} 1_{i} 47∗ 47∗ 22 22 22 22 22 22
3. 2_{−} 2_{+} 2_{−} -1 14 -1 1^{3} 2_{+} 1  1^{2} 1_{k} 47 47 22 3 3 3 22 3 3 3
A_{4} 4 95 1^{2} 1_{k} 47 47 3 3 3 3 3 3

Discriminant class: 2021

∞ 43 47  a_{1} a_{2} a_{3}  Over p = 43  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 31 37 41 53 59 61
4. 2_{−} 3_{k} 1_{k}  2 -20 -21 1^{4} 3_{k} 43 43∗ 1^{2} 1_{k} 47 47 4 4 22 4 2 2  22
D_{4}  121  2 0_{+} ∗ ∗ 2 22 2 4 2  4
5. 2_{−} 1_{k} 1_{k}  0 0 -1 1^{2} 1_{k} 43 43 1^{2} 1_{k} 47 47 2 4 3 4 4 4 3 3
S_{4} 2 2 0_{+} ∗ ∗ 4 3 2 4 2 3 3 4
6. 2_{−} 3_{k} 3_{k}  -1 253 -379 1^{4} 3_{k} 43 43∗ 1^{4} 3_{k} 47∗ 47 2 4 3 4 2 2 3 3
S_{4} 8 16318 2 3 2 2 2 3 3 2
7. 2_{−} 3_{k} 3_{k}  -1 253 -379 1^{4} 3_{k} 43 43∗ 1^{4} 3_{k} 47∗ 47 2 2 3 2 4 4 3 3
S_{4} 8  150  4 3 2 4 2 3 3 4