Quartic Fields Unramified Outside of {∞, 7, 23}

Discriminant class: −23

∞ 7 23 a_{1} a_{2} a_{3} Over p = 7  Over p = 23  2 3 5 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 -13 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{k} 23∗ 23 3 3 4 2 3 4 2 3
S_{4} 2 8 3 4 3 2 3 2 22 4

Discriminant class: −7

∞ 7 23 a_{1} a_{2} a_{3}  Over p = 7  Over p = 23  2 3 5 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} 3_{k} 2_{−} 0 7 0 1^{4} 3_{k} 7∗ 7 1^{2} 1_{i} 23∗ 23∗ 22 4 2 22 4 2 2
D_{4} -28 1^{2} 1_{i} 23∗ 23∗ 4 22 4 22 4 22 4 2

Discriminant class: 1

∞ 7 23 a_{1} a_{2} a_{3} Over p = 7  Over p = 23  2 3 5 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
3. 2_{−} 2_{+} 2_{−} 0 15 0 2^{2} 2_{+} 1 V 1^{2} 1_{i} 23∗ 23∗  22 22 22 22 22 22
V_{4} 3 16 1^{2} 1_{i} 23∗ 23∗ 22 22 22 22 22 22 22 22

Discriminant class: 161

∞ 7 23 a_{1} a_{2} a_{3}  Over p = 7  Over p = 23  2 3 5 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_{−} 3_{i} 1_{i} 2 5 4 1^{4} 3_{i} 7 7∗ 1^{2} 1_{i} 23∗ 23∗ 22 4 22 2 4 22 22
D_{4}  11  4 2 4 2 4 2 4 22