Quartic Fields Unramified Outside of {∞, 23, 41}

Discriminant class: −943

∞ 23 41 a_{1} a_{2} a_{3}  Over p = 23  Over p = 41 2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
1. 1_{k} 1_{i} 3_{+} 2 -19 -20 1^{2} 1_{i} 23∗ 23∗ 1^{4} 3_{+} 41 41  4 2 22 22 4 22 22
D_{4} 6 -64 4 22 2 2 4 22 2

Discriminant class: −23

∞ 23 41 a_{1} a_{2} a_{3}  Over p = 23  Over p = 41 2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
2. 1_{k} 1_{i} 2_{+} 1 -5 -18 1^{2} 1_{i} 23∗ 23∗ 1^{2} 1_{+} 41 41 22 22 2 4 4 22 4 4
D_{4} -4 1^{2} 1_{+} 41 41 22 22 2 2 22 4 2

Discriminant class: 1

∞ 23 41 a_{1} a_{2} a_{3}  Over p = 23  Over p = 41 2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
3. 2_{−} 2_{−} 2_{+} 0 -9 0 1^{2} 1_{i} 23∗ 23∗ 1^{2} 1_{+} 41 41  22 22 22 22 22 22 22
V_{4} 24 256 1^{2} 1_{i} 23∗ 23∗ 1^{2} 1_{+} 41 41 22 22 22 22 22 22

Discriminant class: 41

∞ 23 41 a_{1} a_{2} a_{3}  Over p = 23  Over p = 41 2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
4. 0_{+} 0_{+} 3_{+}  -1 -15 -18  1^{4} 3_{+} 41 41 22 4 22 4 4 4 4 4
C_{4}  -4  4  22 4 4  22
5. 2_{−} 2_{−} 3_{+}  -1 231 474 1^{2} 1_{i} 23∗ 23∗ 1^{4} 3_{+} 41 41 22 4  4 4 4 4 4
C_{4} 32  5900  1^{2} 1_{i} 23∗ 23∗  4  22  4 4
6. 2_{−} 2_{−} 1_{+}  -1 1 14 1^{2} 1_{i} 23∗ 23∗ 1^{2} 1_{+} 41 41 22 2 22 4 4 2 4 4
D_{4} 3  12  1^{2} 1_{i} 23∗ 23∗  2 22 22 22 2 4  22
7. 2_{−} 2_{−} 3_{+}  0 -41 0 1^{2} 1_{i} 23∗ 23∗ 1^{4} 3_{+} 41 41  4 22 2 2 4 2 2
D_{4} 96  656  1^{2} 1_{i} 23∗ 23∗  4 22 22 22 4 2  22