Quartic Fields Ramifying Above {17, 43}

Discriminant class: −731

∞ 17 43 a_{1} a_{2} a_{3}  Over p = 17  Over p = 43  2 3 5 7 11 13 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_{i}  2 -24 -25 1^{4} 3_{+} 17 17 1^{2} 1_{i} 43∗ 43∗ 2 22 22 22 4  2 4
D_{4} 2  -52  22 4 22 4 22 22 22 22
2. 1_{k} 1_{−} 1_{k}  -1 2  0 1^{2} 1_{−} 17∗ 17∗ 1^{2} 1_{k} 43 43 4 3 3 22 4 3 4 4
S_{4}  -1  2 0_{+} ∗  ∗ 2 0_{+} ∗  ∗ 22 2 3 2 3 3 3 3
3. 1_{k} 3_{+} 3_{i}  -1 -91 137 1^{4} 3_{+} 17 17 1^{4} 3_{i} 43∗ 43 4 3 3 22 4 3 4 2
S_{4} 4 2687 22 4 3 4 3 3 3 3
4. 1_{k} 3_{+} 3_{i}  -1 -91 -594 1^{4} 3_{+} 17 17 1^{4} 3_{i} 43∗ 43 2 3 3 22 2 3 2 4
S_{4} 8 -968 4 3 4 3 3 3 3

Discriminant class: −43

∞ 17 43 a_{1} a_{2} a_{3} Over p = 17  Over p = 43 2 3 5 7 11 13 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. 1_{k} 2_{−} 1_{k} 2 4 3 2^{2} 2_{−} 1 V 1^{2} 1_{k} 43 43 2 4 4 4 22  2 22
D_{4} -2 2 0_{+} ∗ ∗ 4 22 4 22 22 4

Discriminant class: 1

∞ 17 43 a_{1} a_{2} a_{3}  Over p = 17  Over p = 43  2 3 5 7 11 13 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_{−} 2_{+} 2_{−} 0 13 0 1^{2} 1_{+} 17 17 1^{2} 1_{i} 43∗ 43∗ 22 22 22 22 22  22 22
V_{4} 6 225 1^{2} 1_{+} 17 17 1^{2} 1_{i} 43∗ 43∗ 22 22 22 22 22

Discriminant class: 17

∞ 17 43 a_{1} a_{2} a_{3}  Over p = 17  Over p = 43  2 3 5 7 11 13 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
7. 2_{−} 3_{+} 2_{−}  1 181 -1 1^{4} 3_{+} 17 17 1^{2} 1_{k} 43 43  4 4 4 4  4
C_{4} 164  8042  1^{2} 1_{k} 43 43 4 4 4 4  22 22 4
8. 2_{−} 1_{−} 2_{+}  -1 7 2 1^{2} 1_{−} 17∗ 17∗ 2^{2} 2_{+} 1  V 22 4 4 4 2  22 2
D_{4}  4  2 0_{+} ∗  ∗  4 2 4 2 22  4
9. 2_{−} 3_{+} 2_{−}  0 -51 0 1^{4} 3_{+} 17 17 1^{2} 1_{i} 43∗ 43∗ 22 2 2 2 4  22 4
D_{4} 24  833  1^{2} 1_{i} 43∗ 43∗ 2 4 2 4 22 22 22 2