Quartic Fields Ramifying Above {3, 43}

Discriminant class: −43

∞ 3 43 a_{1} a_{2} a_{3} Over p = 3  Over p = 43  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 31 37 41 47 53 59 61
1. 1_{k} 4_{+} 3_{i}  1 -48 164 1^{3} 4_{+} ∗  1^{4} 3_{i} 43∗ 43 4 2 2 3 3  4 3
S_{4}  365  2 22 4 3 3 22 22 4

Discriminant class: −3

∞ 3 43 a_{1} a_{2} a_{3}  Over p = 3  Over p = 43 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 31 37 41 47 53 59 61
2. 1_{k} 3_{i} 2_{+} 0 -9 0 1^{4} 3_{i} 3∗ 3 2^{2} 2_{+} 1 V 2 2 22 4 22 4 22 4
D_{4}  -12  2 22 22 4 4 4 4 22
3. 1_{k} 3_{i} 2_{+} -1 -3 9 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  4 4 22 2 3 2 3 4
S_{4}  -3  2 3 22 4 4 4 2 3
4. 1_{k} 5_{i} 2_{+} 1 0 32 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 4 3 4 22 2 3 4
S_{4}  8  2 3 3 4 4 2 2 3

Discriminant class: 1

∞ 3 43 a_{1} a_{2} a_{3} Over p = 3  Over p = 43  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 31 37 41 47 53 59 61
5. 2_{−} 2_{+} 2_{−} -1 -10 -11 2^{2} 2_{+} 1 V 1^{2} 1_{i} 43∗ 43∗ 22 22 22 22  22 22 22
V_{4} 121 1^{2} 1_{i} 43∗ 43∗ 22 22 22 22 22 22 22

Discriminant class: 129

∞ 3 43 a_{1} a_{2} a_{3}  Over p = 3  Over p = 43 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 31 37 41 47 53 59 61
6. 2_{−} 3_{k} 1_{k} 1 6 1 1^{4} 3_{k} 3 3∗ 1^{2} 1_{k} 43 43 22 22 2 4 22 4 2 4
D_{4}  7  2 0_{+} ∗ ∗ 22 22 2 4 4 4 4 2