Quartic Fields Ramifying Above {5, 29}

Discriminant class: 1

∞ 5 29 a_{1} a_{2} a_{3} Over p = 5  Over p = 29 2 3 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. 0_{+} 2_{+} 2_{+} 0 -17 0 1^{2} 1_{+} 5 5 1^{2} 1_{+} 29 29 22 22 22 22 22 22 22 22
V_{4} 2 36 1^{2} 1_{+} 5 5 1^{2} 1_{+} 29 29 22 22 22 22 22 22 22

Discriminant class: 5

∞ 5 29 a_{1} a_{2} a_{3}  Over p = 5  Over p = 29  2 3 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_{−} 3_{−} 2_{+} -1 36 -36 1^{4} 3_{−} 5 5 1^{2} 1_{−} 29∗ 29∗ 4 4 4 22 4 4  4
C_{4} 4 281 1^{2} 1_{−} 29∗ 29∗ 22 4 22 4 4 4 22 22
3. 0_{+} 1_{−} 2_{−} 1 -5 1 1^{2} 1_{−} 5∗ 5∗ 2^{2} 2_{−} 1  V 4 4 2 22 2 4 22 2
D_{4} 1 2 0_{+} ∗ ∗ 22 4 22 4 4 2 22 22
4. 2_{−} 3_{−} 2_{+} 2 14 13 1^{4} 3_{−} 5 5 1^{2} 1_{+} 29 29 2 2 4 22 4 2 22 4
D_{4} 8 6 1^{2} 1_{+} 29 29 22 2 22 2 2 4 22

Discriminant class: 29

∞ 5 29 a_{1} a_{2} a_{3}  Over p = 5  Over p = 29  2 3 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
5. 2_{−} 2_{+} 3_{−} -1 33 -107 1^{2} 1_{−} 5∗ 5∗ 1^{4} 3_{−} 29 29 4 4 22 4  4 4 22
C_{4} 4 139 1^{2} 1_{−} 5∗ 5∗ 4 4 4 4 4 22 4
6. 0_{+} 2_{−} 1_{−} -1 -3  1 2^{2} 2_{−} 1 V 1^{2} 1_{−} 29∗ 29∗ 4 4 22 2 22 4 2 22
D_{4} 1 2 0_{+} ∗ ∗ 2 4 2 4 4 22 22 2
7. 2_{−} 2_{+} 3_{−} 2 16 15 1^{2} 1_{+} 5 5 1^{4} 3_{−} 29 29 2 2 22 4 22 2 4 22
D_{4} 24 20 1^{2} 1_{+} 5 5 4 2 4 2 2 22 22 4

Discriminant class: 145

∞ 5 29 a_{1} a_{2} a_{3} Over p = 5  Over p = 29 2 3 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
8. 0_{+} 3_{−} 3_{−} 1 -54 -9 1^{4} 3_{−} 5 5 1^{4} 3_{−} 29 29 22  4 4 4 22 4 4
C_{4} 4  81  4  4  4 22 4
9. 0_{+} 3_{−} 3_{−} -1 -54 -136 1^{4} 3_{−} 5 5 1^{4} 3_{−} 29 29  22 4 4 4  4 4
C_{4} 4  -64  4 22 4 22 22 4 22 4
10. 2_{−} 3_{−} 1_{+} -1 6 -1 1^{4} 3_{−} 5 5 1^{2} 1_{+} 29 29 22 22 4 2 4 22 2 4
D_{4} 2 11 2 22 2 22 22 4 2
11. 2_{−} 1_{+} 3_{−} 1 4 -9 1^{2} 1_{+} 5 5 1^{4} 3_{−} 29 29 22 22 2 4 2 22 4 2
D_{4} 6  23  4 22 4 22 22 2 22 4