Quartic Fields Unramified Outside of {∞, 19, 43}

Discriminant class: −19

∞ 19 43 a_{1} a_{2} a_{3}  Over p = 19  Over p = 43 2 3 5 7 11 13 17 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_{i} 2_{+} -2 30 -29 1^{4} 3_{i} 19∗ 19 2^{2} 2_{+} 1 V 2 2 22 22 22 4  22
D_{4} 5 6 2 4 2 4 4 4 22

Discriminant class: 1

∞ 19 43 a_{1} a_{2} a_{3} Over p = 19  Over p = 43  2 3 5 7 11 13 17 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 31 0 2^{2} 2_{+} 1 V 1^{2} 1_{i} 43∗ 43∗ 22 22 22 22  22
V_{4} 5 36 1^{2} 1_{i} 43∗ 43∗ 22 22 22 22 22 22 22

Discriminant class: 817

∞ 19 43 a_{1} a_{2} a_{3}  Over p = 19  Over p = 43 2 3 5 7 11 13 17 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. 2_{−} 3_{k} 1_{k} -1 17 -6 1^{4} 3_{k} 19 19∗ 1^{2} 1_{k} 43 43 22 22 2 2 22 4  22
D_{4}  36  2 0_{+} ∗ ∗ 22 4 22 4  4 4 2