Quartic Fields Ramifying Above {11, 41}

Discriminant class: −451

∞ 11 41 a_{1} a_{2} a_{3}  Over p = 11  Over p = 41  2 3 5 7 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_{−}  0 0 -11 1^{4} 3_{k} 11 11∗ 1^{2} 1_{−} 41∗ 41∗ 4 4 3 3 22 3 3 3
S_{4} 11 3 3 3 4 4 2 3 4
2. 1_{k} 1_{k} 3_{−}  0 0 -41 1^{2} 1_{k} 11 11 1^{4} 3_{−} 41∗ 41∗ 4 2 3 3 22 3 3 3
S_{4} 4 -123 3 3 3 2 4 4 3 2
3. 1_{k} 3_{k} 3_{−} -1 -56 -141 1^{4} 3_{k} 11 11∗ 1^{4} 3_{−} 41∗ 41∗ 2 4 3 3 22 3 3 3
S_{4} 8 37 3 3 3 4 2 4 3 4

Discriminant class: 1

∞ 11 41 a_{1} a_{2} a_{3} Over p = 11  Over p = 41 2 3 5 7 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. 2_{−} 2_{+} 2_{−} 0 -15 0 2^{2} 2_{+} 1 V 2^{2} 2_{−} 1 V 22 22  22 22 22 22
V_{4} 3 169 22 22 22 22 22

Discriminant class: 41

∞ 11 41 a_{1} a_{2} a_{3} Over p = 11  Over p = 41 2 3 5 7 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
5. 2_{−} 2_{−} 3_{+}  1 108 -228 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{+} 41 41  4 22 4 4 4 4
C_{4} 74  1472  4  4 4