Quartic Fields Ramifying Above {11, 47}

Discriminant class: −11

∞ 11 47 a_{1} a_{2} a_{3}  Over p = 11  Over p = 47 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 41 43 53 59 61
1. 1_{k} 3_{i} 2_{+} 0 -11 0 1^{4} 3_{i} 11∗ 11 2^{2} 2_{+} 1 V 4 22 22 4 2 4 2 22
D_{4} -99 2 22 2 2 22 22 4

Discriminant class: 1

∞ 11 47 a_{1} a_{2} a_{3} Over p = 11  Over p = 47  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 41 43 53 59 61
2. 2_{−} 2_{+} 2_{−} 0 29 0 2^{2} 2_{+} 1 V 1^{2} 1_{i} 47∗ 47∗ 22  22 22 22 22 22 22
V_{4} 5 81 1^{2} 1_{i} 47∗ 47∗ 22 22 22 22 22

Discriminant class: 517

∞ 11 47 a_{1} a_{2} a_{3}  Over p = 11  Over p = 47 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 41 43 53 59 61
3. 2_{−} 3_{k} 1_{k} 2 -4 -5 1^{4} 3_{k} 11 11∗ 1^{2} 1_{k} 47 47 4 22 2 4 22 4 22 2
D_{4}  31  2 0_{+} ∗ ∗ 22 2  22 22 22 22 4