Quartic Fields Unramified Outside of {∞, 11, 19}

Discriminant class: −19

∞ 11 19 a_{1} a_{2} a_{3} Over p = 11  Over p = 19  2 3 5 7 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} 2_{+} 3_{i}  2 -8 -9 2^{2} 2_{+} 1 V 1^{4} 3_{i} 19∗ 19 2 4 22 22 2 22  2
D_{4}  -32  4 4 2 22  4 4 22

Discriminant class: 1

∞ 11 19 a_{1} a_{2} a_{3}  Over p = 11  Over p = 19 2 3 5 7 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. 2_{−} 2_{−} 2_{+} 0 15 0 1^{2} 1_{i} 11∗ 11∗ 2^{2} 2_{+} 1 V 22 22  22 22 22  22
V_{4}  4  1^{2} 1_{i} 11∗ 11∗  22 22 22 22  22 22 22
3. 2_{−} 2_{−} 2_{+} -1 -9 6 1^{2} 1_{k} 11 11 1^{3} 2_{+} 1  3 3 3 22 3 3 3 3
A_{4} 4 25 1^{2} 1_{k} 11 11 22 22 3 3 3 3 3 3

Discriminant class: 209

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