Quartic Fields Unramified Outside of {∞, 41, 43}

Discriminant class: −1763

∞ 41 43 a_{1} a_{2} a_{3}  Over p = 41  Over p = 43  2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 23 29 31 37 47 53 59 61
1. 1_{k} 3_{+} 1_{i}  -1 -15 146 1^{4} 3_{+} 41 41 1^{2} 1_{i} 43∗ 43∗ 2 22 2 22 4 4 4 22
D_{4} 2 -496 22 2 4 4 22 2
2. 1_{k} 3_{+} 3_{i}  -1 -220 1212 1^{4} 3_{+} 41 41 1^{4} 3_{i} 43∗ 43 2 3 2 3 4 2 4
S_{4} 8 2128 3 22 3 2 4 4 3 4

Discriminant class: −43

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

Discriminant class: 1

∞ 41 43 a_{1} a_{2} a_{3}  Over p = 41  Over p = 43  2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 23 29 31 37 47 53 59 61
4. 2_{−} 2_{+} 2_{−} 0 1 0 1^{2} 1_{+} 41 41 1^{2} 1_{i} 43∗ 43∗ 22 22 22 22 22 22 22 22
V_{4} 6 441 1^{2} 1_{+} 41 41 1^{2} 1_{i} 43∗ 43∗ 22 22 22 22 22
5. 2_{−} 2_{−} 2_{+} 0 -17 82 2^{2} 2_{−} 1 V 1^{3} 2_{+} 1  22 3 3 3 22 3 3 3
A_{4} 2 431 3 3 3 3 22 3 22 3

Discriminant class: 41

∞ 41 43  a_{1} a_{2} a_{3}  Over p = 41  Over p = 43  2 3 5 7 11 13 17 19
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 23 29 31 37 47 53 59 61
6. 0_{+} 3_{+} 0_{+}  -1 -15 -18 1^{4} 3_{+} 41 41 2 0_{+} ∗  ∗ 22 4 22 4 4 4 4 4
C_{4}  -4  2 0_{+} ∗  ∗  4  4 4  22
7. 2_{−} 3_{+} 2_{−}  -1 436 884 1^{4} 3_{+} 41 41 1^{2} 1_{k} 43 43  4  4 4 4 4 4
C_{4} 580  19840  1^{2} 1_{k} 43 43  4  22 4 4
8. 2_{−} 1_{−} 2_{+}  -1 3 4 1^{2} 1_{−} 41∗ 41∗ 2^{2} 2_{+} 1  V 22 4 22 4 2 2 2 4
D_{4}  16  2 0_{+} ∗  ∗  4  22 2 2 22 22
9. 2_{−} 3_{+} 2_{−}  2 -19 -20 1^{4} 3_{+} 41 41 1^{2} 1_{i} 43∗ 43∗ 22 2 22 2 4 4 4 2
D_{4} 24  1863  1^{2} 1_{i} 43∗ 43∗  2  22 4 4 22 22