Quartic Fields Unramified Outside of {∞, 11, 23}

Discriminant class: −23

∞ 11 23 a_{1} a_{2} a_{3} Over p = 11  Over p = 23  2 3 5 7 13 17 19 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_{k} -1 9 -36 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{k} 23∗ 23 3 3 4 4 3 2 4 3
S_{4} 2 31 3 2 3 4 3 2 22 2

Discriminant class: −11

∞ 11 23 a_{1} a_{2} a_{3}  Over p = 11  Over p = 23 2 3 5 7 13 17 19 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_{+} 2 -4 -5 1^{4} 3_{i} 11∗ 11 2^{2} 2_{+} 1 V 4  22 2 4 2 2 4
D_{4} -57 22 22 4 2 22 22 2

Discriminant class: 1

∞ 11 23 a_{1} a_{2} a_{3} Over p = 11  Over p = 23  2 3 5 7 13 17 19 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 17 0 2^{2} 2_{+} 1 V 1^{2} 1_{i} 23∗ 23∗ 22  22 22 22 22 22 22
V_{4} 3 9 1^{2} 1_{i} 23∗ 23∗ 22 22 22 22 22

Discriminant class: 253

∞ 11 23 a_{1} a_{2} a_{3}  Over p = 11  Over p = 23 2 3 5 7 13 17 19 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} -1 -1 13 1^{4} 3_{k} 11 11∗ 1^{2} 1_{k} 23 23 4  2 22 4 22 22 4
D_{4}  15  2 0_{+} ∗ ∗ 22 2 4 22  2 22 22