Quartic Fields Ramifying Above {11, 29}

Discriminant class: −11

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

Discriminant class: 1

∞ 11 29 a_{1} a_{2} a_{3} Over p = 11  Over p = 29 2 3 5 7 13 17 19 23
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
2. 2_{−} 2_{+} 2_{−} 0 -9 0 2^{2} 2_{+} 1 V 2^{2} 2_{−} 1 V 22 22  22 22 22 22
V_{4} 5 100 22 22 22 22 22 22

Discriminant class: 29

∞ 11 29 a_{1} a_{2} a_{3} Over p = 11  Over p = 29 2 3 5 7 13 17 19 23
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
3. 0_{+} 2_{−} 3_{−} 1 -83 -241 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{−} 29 29 4 4 22 22  4 4
C_{4}  23  4 4 4 4 4  4