Quartic Fields Unramified Outside of {∞, 3, 23}

Discriminant class: −23

∞ 3 23 a_{1} a_{2} a_{3}  Over p = 3  Over p = 23  2 5 7 11 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 -14 33 1^{2} 1_{i} 3∗ 3∗ 1^{4} 3_{k} 23∗ 23 22 2 4 2 22 2 4 22
D_{4} -15 1^{2} 1_{i} 3∗ 3∗ 22 4 22 4 22 2 22 4

Discriminant class: −3

∞ 3 23 a_{1} a_{2} a_{3}  Over p = 3  Over p = 23  2 5 7 11 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} 5_{i} 2_{+} -1 -9 19 1^{3} 5_{i} 3∗  1^{2} 1_{i} 23∗ 23∗ 4 4 3 2 3 4 3 4
S_{4}  -7  1^{2} 1_{k} 23 23 3 3 2 3 2 4 4
3. 1_{k} 5_{i} 2_{+} -1 9 33 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  4 2 3 4 22 4 3 4
S_{4}  -15  3 3 4  2 2 2 22

Discriminant class: 1

∞ 3 23 a_{1} a_{2} a_{3}  Over p = 3  Over p = 23 2 5 7 11 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_{−} 2_{−} 2_{+} -1 -5 -6 1^{2} 1_{i} 3∗ 3∗ 2^{2} 2_{+} 1 V 22 22 22 22  22 22 22
V_{4} 3 36 1^{2} 1_{i} 3∗ 3∗ 22 22 22 22 22 22 22

Discriminant class: 69

∞ 3 23 a_{1} a_{2} a_{3}  Over p = 3  Over p = 23  2 5 7 11 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
5. 2_{−} 1_{i} 3_{i} -1 9 10 1^{2} 1_{i} 3∗ 3∗ 1^{4} 3_{i} 23 23∗ 2 22 4 22 22 22 4 2
D_{4} 3 8 22 4 2 4 2 22 2 4
6. 2_{−} 3_{i} 3_{i} -1 9 -13 1^{3} 3_{i} 3∗  1^{4} 3_{i} 23 23∗ 4 3 2 3 3 3 4 4
S_{4} 2  31  3 2 4 4 4 3 2 2