Quartic Fields Unramified Outside of {∞, 7, 43}

Discriminant class: −7

∞ 7 43 a_{1} a_{2} a_{3}  Over p = 7  Over p = 43  2 3 5 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} 3_{k} 2_{−} 0 -7 0 1^{4} 3_{k} 7∗ 7 1^{2} 1_{i} 43∗ 43∗ 22 2 2 22 4 4 2
D_{4} -63 1^{2} 1_{i} 43∗ 43∗ 22 4 22 4 4 22 4 2

Discriminant class: 1

∞ 7 43 a_{1} a_{2} a_{3} Over p = 7  Over p = 43  2 3 5 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 25 0 2^{2} 2_{+} 1 V 1^{2} 1_{i} 43∗ 43∗ 22 22 22  22 22 22
V_{4} 81 1^{2} 1_{i} 43∗ 43∗ 22 22 22 22 22 22 22
3. 2_{−} 2_{+} 2_{−} 1 -7 -9 1^{3} 2_{+} 1  1^{2} 1_{i} 43∗ 43∗ 3 3 3 3 22 3 3 3
A_{4} 4 38 1^{2} 1_{i} 43∗ 43∗ 22 3 3 22 3 3 3 3

Discriminant class: 301

∞ 7 43 a_{1} a_{2} a_{3}  Over p = 7  Over p = 43  2 3 5 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
4. 2_{−} 3_{i} 1_{i} 2 -2 -3 1^{4} 3_{i} 7 7∗ 1^{2} 1_{i} 43∗ 43∗ 2 22 22 22 4 4 22
D_{4}  18  2 4 2 4 4 22 4 22