Quartic Fields Ramifying Above {3, 19}

Discriminant class: −3

∞ 3 19 a_{1} a_{2} a_{3}  Over p = 3  Over p = 19 2 5 7 11 13 17 23 29
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 1_{k} 3_{i} 2_{+} 0 3 0 1^{4} 3_{i} 3∗ 3 2^{2} 2_{+} 1 V 2 4 22 4 22 4 4 2
D_{4}  -12  22 22 2  4 2 2 22
2. 1_{k} 5_{i} 2_{+} -1 6 8 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  4 2 3 4 3 4 4 4
S_{4}  -5  22 22 2  2 4 4 3
3. 1_{k} 5_{i} 2_{+} 0 -6 -11 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  4 4 22 2 22 2 4 4
S_{4}  -3  3 3 4 3 2 2 2 3

Discriminant class: 1

∞ 3 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 19  2 5 7 11 13 17 23 29
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
4. 2_{−} 2_{+} 2_{−} -1 -4 -5 2^{2} 2_{+} 1 V 1^{2} 1_{i} 19∗ 19∗ 22 22  22 22 22 22 22
V_{4} 25 1^{2} 1_{i} 19∗ 19∗ 22 22 22 22 22 22
5. 2_{−} 4_{+} 2_{−} -1 3 1 1^{3} 4_{+} 1  1^{2} 1_{k} 19 19 3 3 3 3 3 22 3 3
A_{4} 4 20 1^{2} 1_{k} 19 19 3 22 3 3 3 22 3 3

Discriminant class: 57

∞ 3 19 a_{1} a_{2} a_{3}  Over p = 3  Over p = 19 2 5 7 11 13 17 23 29
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
6. 2_{−} 3_{k} 1_{k} -2 3 -2 1^{4} 3_{k} 3 3∗ 1^{2} 1_{k} 19 19 22 4 22 4 2 4 4 22
D_{4}  4  2 0_{+} ∗ ∗ 2 2 22  4 22 22 22