Quartic Fields Ramifying Above {5, 43}

Discriminant class: −43

∞ 5 43 a_{1} a_{2} a_{3} Over p = 5  Over p = 43  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 31 37 41 47 53 59 61
1. 1_{k} 2_{+} 3_{i} -1 -5 -35 1^{3} 2_{+} ∗  1^{4} 3_{i} 43∗ 43 4 2 2 3 3 22 2 3
S_{4} -65 4 3 2 3 3 3 3 4

Discriminant class: 1

∞ 5 43 a_{1} a_{2} a_{3} Over p = 5  Over p = 43 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 31 37 41 47 53 59 61
2. 2_{−} 2_{−} 2_{+} 0 19 0 2^{2} 2_{−} 1 V 2^{2} 2_{+} 1 V 22 22 22  22 22 22 22
V_{4} 7 144 22 22 22 22 22

Discriminant class: 5

∞ 5 43 a_{1} a_{2} a_{3} Over p = 5  Over p = 43 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 31 37 41 47 53 59 61
3. 0_{+} 3_{−} 2_{−} 1 -54 -54 1^{4} 3_{−} 5 5 2^{2} 2_{−} ∗ ∗ 4 4 4  4 4  4
C_{4}  551  4  4 4 22 22