Quartic Fields Ramifying Above {29, 47}

Discriminant class: −1363

∞ 29 47 a_{1} a_{2} a_{3}  Over p = 29  Over p = 47  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 31 37 41 43 53 59 61
1. 1_{k} 3_{+} 3_{i}  -1 -170 -426 1^{4} 3_{+} 29∗ 29∗ 1^{4} 3_{i} 47 47∗ 4 4 4 3 3 2 2 22
S_{4} 4 2923 4 3 2 3 3 3 3 4

Discriminant class: −47

∞ 29 47 a_{1} a_{2} a_{3} Over p = 29  Over p = 47  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 31 37 41 43 53 59 61
2. 1_{k} 2_{+} 1_{i} 0 -7 -7 1^{3} 2_{+} ∗  1^{2} 1_{i} 47∗ 47∗ 3 22 2 22 4 4 3 4
S_{4} 8 4 2 3 4 2 3 22 3

Discriminant class: 1

∞ 29 47 a_{1} a_{2} a_{3} Over p = 29  Over p = 47 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 31 37 41 43 53 59 61
3. 2_{−} 2_{−} 2_{+} 0 9 0 2^{2} 2_{−} 1 V 2^{2} 2_{+} 1 V 22 22 22  22 22 22 22
V_{4} 15 361 22 22 22 22 22 22

Discriminant class: 29

∞ 29 47 a_{1} a_{2}  a_{3}  Over p = 29  Over p = 47 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 31 37 41 43 53 59 61
4. 0_{+} 3_{−} 2_{−}  1 -344 -1024 1^{4} 3_{−} 29 29 2^{2} 2_{−} ∗ ∗ 4 4  4  4 4
C_{4}  3155  22 4 4 4 4  4