Quartic Fields Ramifying Above {13, 43}

Discriminant class: −559

∞ 13 43 a_{1} a_{2} a_{3}  Over p = 13  Over p = 43 2 3 5 7 11 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} 3_{−} 1_{k} 1 -11 22 1^{4} 3_{−} 13 13 1^{2} 1_{k} 43 43 22 2 22 22 4 22 22
D_{4} -36 2 0_{+} ∗ ∗ 2 4 22 4 4 22 4 2

Discriminant class: −43

∞ 13 43 a_{1} a_{2} a_{3}  Over p = 13  Over p = 43  2 3 5 7 11 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} 2_{+} 1_{i}  2 0 -1 1^{2} 1_{+} 13 13 1^{2} 1_{i} 43∗ 43∗ 4 2 4 4 22  4 22
D_{4}  -3  1^{2} 1_{+} 13 13  2 22 4 22 22 22 22 2
3. 1_{k} 2_{+} 3_{i}  -1 -5 223 1^{3} 2_{+} 1  1^{4} 3_{i} 43∗ 43 4 4 4 4 3 3 4 3
S_{4}  -151  2 3 4 22 22 22 3 4

Discriminant class: 1

∞ 13 43 a_{1} a_{2} a_{3}  Over p = 13  Over p = 43  2 3 5 7 11 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
4. 2_{−} 2_{+} 2_{−} 0 15 0 1^{2} 1_{+} 13 13 1^{2} 1_{i} 43∗ 43∗ 22 22 22 22 22  22
V_{4} 8 196 1^{2} 1_{+} 13 13 1^{2} 1_{i} 43∗ 43∗ 22 22 22 22 22 22 22

Discriminant class: 13

∞ 13 43 a_{1} a_{2} a_{3}  Over p = 13  Over p = 43  2 3 5 7 11 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. 0_{+} 3_{−} 2_{−}  -1 -141 -139 1^{4} 3_{−} 13 13 1^{2} 1_{k} 43 43 4 22 4 4 4 22 4 22
C_{4}  1433  1^{2} 1_{k} 43 43 22 4 4 4 4  4 22
6. 2_{−} 1_{+} 2_{−}  0 -3  0 1^{2} 1_{+} 13 13 1^{2} 1_{i} 43∗ 43∗ 4 22 4 4 2  4 22
D_{4}  13  1^{2} 1_{i} 43∗ 43∗ 22 2 4 2 2 22 2 22
7. 2_{−} 3_{−} 2_{+}  2 -18 -19 1^{4} 3_{−} 13 13 2^{2} 2_{+} 1  V 2 22 2 2 4 22 2
D_{4} 16  230  22 4 2 4 4 22 4 22