Quartic Fields Unramified Outside of {∞, 19, 47}

Discriminant class: −19

∞ 19 47 a_{1} a_{2} a_{3}  Over p = 19  Over p = 47 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 43 53 59 61
1. 1_{k} 3_{i} 2_{+}  2 -8 -9 1^{4} 3_{i} 19∗ 19 2^{2} 2_{+} 1 V 4 4 22  22 2  22
D_{4}  -203  2 2 4 2 22 4 4

Discriminant class: 1

∞ 19 47 a_{1} a_{2} a_{3} Over p = 19  Over p = 47  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 43 53 59 61
2. 2_{−} 2_{+} 2_{−} 0 33 0 2^{2} 2_{+} 1 V 1^{2} 1_{i} 47∗ 47∗ 22 22 22  22 22  22
V_{4} 5  49  1^{2} 1_{i} 47∗ 47∗ 22 22 22 22 22 22 22

Discriminant class: 893

∞ 19 47 a_{1} a_{2} a_{3}  Over p = 19  Over p = 47 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 43 53 59 61
3. 2_{−} 3_{k} 1_{k} 1 -2 -32 1^{4} 3_{k} 19 19∗ 1^{2} 1_{k} 47 47 4 4 2  2 22  2
D_{4}  55  2 0_{+} ∗ ∗ 22 22 4 22 2 4 4