Quartic Fields Unramified Outside of {∞, 7, 19}

Discriminant class: −19

∞ 7 19 a_{1} a_{2} a_{3} Over p = 7  Over p = 19  2 3 5 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} 2_{+} 3_{i}  2 -8 -9 2^{2} 2_{+} 1 V 1^{4} 3_{i} 19∗ 19 4 2 22 22 2 22  4
D_{4}  -13  2 4 2  22 4 2 22

Discriminant class: 1

∞ 7 19 a_{1} a_{2} a_{3}  Over p = 7  Over p = 19 2 3 5 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. 2_{−} 2_{−} 2_{+} 0 13 0 1^{2} 1_{i} 7∗ 7∗ 2^{2} 2_{+} 1 V 22 22 22  22 22  22
V_{4}  9  1^{2} 1_{i} 7∗ 7∗  22 22 22  22 22 22 22
3. 2_{−} 2_{−} 2_{+} 0 3 -7 1^{2} 1_{k} 7 7 1^{3} 2_{+} 1  3 3 3 22 3 3 3 3
A_{4} 4 4 1^{2} 1_{k} 7 7 22 22 3 3 3 3 3 3

Discriminant class: 133

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