Quartic Fields Ramifying Above {3, 29}

Discriminant class: −87

∞ 3 29 a_{1} a_{2} a_{3}  Over p = 3  Over p = 29  2 5 7 11 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} 3_{+} -1 4 9 1^{2} 1_{i} 3∗ 3∗ 1^{4} 3_{+} 29∗ 29∗ 3 4 3 3 3 3 2 4
S_{4} 2 -6 2 0_{+} ∗ ∗ 2 2 22 4 3 2 4 4

Discriminant class: 1

∞ 3 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 29 2 5 7 11 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_{−} -1 8 7 2^{2} 2_{+} 1 V 2^{2} 2_{−} 1 V 22 22  22  22 22 22
V_{4} 3  49  22 22 22 22 22 22 22 22

Discriminant class: 29

∞ 3 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 29  2 5 7 11 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 -25 -67 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{−} 29 29 4  4 22 4 4 22
C_{4} -35 4 4 4 4 4 22 22 4
4. 2_{−} 4_{+} 1_{−} -1 3 -3 1^{3} 4_{+} ∗  1^{2} 1_{−} 29∗ 29∗ 4 3 3 2 22 2 4 3
S_{4} 3 2 0_{+} ∗ ∗ 4 2 4 4 4 3 3 4
5. 2_{−} 4_{+} 3_{−} 0 0 87 1^{3} 4_{+} ∗  1^{4} 3_{−} 29 29 4 3 3 2 22 4 4 3
S_{4} 8 261 2 4 2 4 2 3 3 2
6. 2_{−} 4_{+} 3_{−} -1 33 -20 1^{3} 4_{+} ∗  1^{4} 3_{−} 29 29 2 3 3 2 22 4 2 3
S_{4} 8  284  4 4 4 2 4 3 3 4