Quartic Fields Unramified Outside of {∞, 19, 23}

Discriminant class: −23

∞ 19 23 a_{1} a_{2} a_{3} Over p = 19  Over p = 23 2 3 5 7 11 13 17 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_{−} 1_{k} -1 -3 4 2^{2} 2_{−} ∗ ∗ 1^{2} 1_{k} 23 23 3 3 4 4 2 3 2 3
S_{4} -3 2 0_{+} ∗ ∗ 3 4 3 4 3 4 22 4

Discriminant class: −19

∞ 19 23 a_{1} a_{2} a_{3}  Over p = 19  Over p = 23 2 3 5 7 11 13 17 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} 3_{i} 2_{+} 0 -19 0 1^{4} 3_{i} 19∗ 19 2^{2} 2_{+} 1 V 4 4 22 22 22 4 22 4
D_{4} -19 4 2 4 22 22 2 4 22

Discriminant class: 1

∞ 19 23 a_{1} a_{2} a_{3} Over p = 19  Over p = 23  2 3 5 7 11 13 17 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 21 0 2^{2} 2_{+} 1 V 1^{2} 1_{i} 23∗ 23∗ 22 22 22 22 22 22 22 22
V_{4} 3 1 1^{2} 1_{i} 23∗ 23∗ 22 22 22 22 22 22 22

Discriminant class: 437

∞ 19 23 a_{1} a_{2} a_{3}  Over p = 19  Over p = 23 2 3 5 7 11 13 17 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_{−} 3_{k} 1_{k} 2 -8 -9 1^{4} 3_{k} 19 19∗ 1^{2} 1_{k} 23 23 4 4 2 2 2 4 2 4
D_{4}  25  2 0_{+} ∗ ∗ 4 22 4 2 22 22 4 2