Quartic Fields Ramifying Above {5, 19}

Discriminant class: −95

∞ 5 19 a_{1} a_{2} a_{3} Over p = 5  Over p = 19 2 3 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_{−} 1_{k} -1 1 -6 1^{4} 3_{−} 5 5 1^{2} 1_{k} 19 19 22 22 4 22 22 4 4 2
D_{4} -4 2 0_{+} ∗ ∗ 2 22 2 4 4 22 2 22

Discriminant class: −19

∞ 5 19 a_{1} a_{2} a_{3} Over p = 5  Over p = 19  2 3 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
2. 1_{k} 2_{+} 1_{i} -2 2 -1 1^{2} 1_{+} 5 5 1^{2} 1_{i} 19∗ 19∗ 4 4 22 22 4 22 22 2
D_{4} -1 1^{2} 1_{+} 5 5 2 4 2 22 22 4 2 22

Discriminant class: 1

∞ 5 19 a_{1} a_{2} a_{3} Over p = 5  Over p = 19  2 3 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
3. 2_{−} 2_{+} 2_{−} 0 7 0 1^{2} 1_{+} 5 5 1^{2} 1_{i} 19∗ 19∗ 22 22 22  22 22 22 22
V_{4} 4 36 1^{2} 1_{+} 5 5 1^{2} 1_{i} 19∗ 19∗ 22 22 22 22 22 22 22

Discriminant class: 5

∞ 5 19 a_{1} a_{2} a_{3} Over p = 5  Over p = 19  2 3 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. 0_{+} 3_{−} 2_{−} 1 -24 -24 1^{4} 3_{−} 5 5 1^{2} 1_{k} 19 19 4 4 4  4 4 4
C_{4} 101 1^{2} 1_{k} 19 19 22 4 22 4 4 4
5. 2_{−} 1_{+} 2_{−} -1 3 1 1^{2} 1_{+} 5 5 1^{2} 1_{i} 19∗ 19∗ 4 4 2 22 4 2 2 22
D_{4} 1 1^{2} 1_{i} 19∗ 19∗ 22 4 22 2 2 4 22 22
6. 2_{−} 3_{−} 2_{+} 2 4 3 1^{4} 3_{−} 5 5 2^{2} 2_{+} 1  V 2 2 4 22 2 4 4 22
D_{4} 8  26  22 2 22 4 4 2 22 22