Quartic Fields Ramifying Above {5, 47}

Discriminant class: 1

∞ 5 47 a_{1} a_{2} a_{3} Over p = 5  Over p = 47 2 3 7 11 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. 2_{−} 2_{−} 2_{+} 0 21 0 2^{2} 2_{−} 1 V 2^{2} 2_{+} 1 V 22 22 22 22 22 22 22 22
V_{4} 5 169 22 22 22 22 22 22

Discriminant class: 5

∞ 5 47 a_{1} a_{2} a_{3}  Over p = 5  Over p = 47 2 3 7 11 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. 0_{+} 3_{−} 2_{−} 1 -59 -59 1^{4} 3_{−} 5 5 2^{2} 2_{−} ∗ ∗ 4 4 4 22 4 4  4
C_{4} 661 22 4 22 4 4 22
3. 2_{−} 1_{−} 2_{+} 2 -2 -1 1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} ∗  4 4 2 3 4 4 3 2
S_{4} 7 2 0_{+} ∗ ∗ 3 3 2 3 4 4 3 3
4. 2_{−} 3_{−} 2_{+} -1 1 14 1^{4} 3_{−} 5 5 1^{3} 2_{+} ∗  2 4 2 3 2 4 3 4
S_{4} 8 16 3 3 2 3 4 2 3 3
5. 2_{−} 3_{−} 2_{+} -1 -14 4 1^{4} 3_{−} 5 5 1^{3} 2_{+} ∗  4 2 2 3 4 2 3 4
S_{4} 8  61  3 3 2 3 2 4 3 3