Quartic Fields Ramifying Above {3, 41}

Discriminant class: −123

∞ 3 41 a_{1} a_{2} a_{3} Over p = 3  Over p = 41  2 5 7 11 13 17 19 23
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} 3_{k} 1_{−} -1 0 -2 1^{3} 3_{k} 3  1^{2} 1_{−} 41∗ 41∗ 4 4 4 3 4 22 4 2
S_{4} -1 3 3 3 3 3 3 4 3
2. 1_{k} 3_{k} 3_{−} -1 -15 23 1^{3} 3_{k} 3  1^{4} 3_{−} 41∗ 41∗ 4 2 4 3 2 22 4 4
S_{4} 4  37  3 3 3 3 3 3 2 3
3. 1_{k} 3_{k} 3_{−}  1 -15 18 1^{3} 3_{k} 3  1^{4} 3_{−} 41∗ 41∗ 2 4 2 3 4 22 2 4
S_{4} 4  -168  3 3 3 3 3 3 4 3

Discriminant class: −3

∞ 3 41 a_{1} a_{2} a_{3}  Over p = 3  Over p = 41 2 5 7 11 13 17 19 23
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. 1_{k} 5_{i} 2_{+} -1 9 9 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  4 4 3 2 22 2 22 2
S_{4}  -9  4 3  3 2 2 2 22
5. 1_{k} 5_{i} 2_{+} -1 -18 -44 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 4 3 4 3 4 3 4
S_{4}  -16  4  3 3 4 4 2 22

Discriminant class: 1

∞ 3 41 a_{1} a_{2} a_{3} Over p = 3  Over p = 41 2 5 7 11 13 17 19 23
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
6. 2_{−} 2_{+} 2_{−} -1 11 10 2^{2} 2_{+} 1 V 2^{2} 2_{−} 1 V 22 22 22 22 22 22 22 22
V_{4} 100 22 22 22 22

Discriminant class: 41

∞ 3 41 a_{1} a_{2} a_{3} Over p = 3  Over p = 41 2 5 7 11 13 17 19 23
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
7. 2_{−} 2_{−} 3_{+} 1 26 -64 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{+} 41 41  4 4 4 4 4 22
C_{4} 34  160  4  22 4 4 22 22