Quartic Fields Ramifying Above {17, 47}

Discriminant class: −799

∞ 17 47 a_{1} a_{2} a_{3}  Over p = 17  Over p = 47  2 3 5 7 11 13 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. 1_{k} 3_{+} 1_{i}  2 -7 -8 1^{4} 3_{+} 17 17 1^{2} 1_{i} 47∗ 47∗  4 22 4 22 2 2 22
D_{4} 2  -52  22 22 4 22 2 22  4

Discriminant class: −47

∞ 17 47 a_{1} a_{2} a_{3}  Over p = 17  Over p = 47  2 3 5 7 11 13 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. 1_{k} 2_{+} 1_{i} -2 9 -8 1^{2} 1_{+} 17 17 1^{2} 1_{i} 47∗ 47∗ 22 22 4 22 4 2 2 4
D_{4} -1 1^{2} 1_{+} 17 17 4 4 22 4 2 22 22

Discriminant class: 1

∞ 17 47 a_{1} a_{2} a_{3}  Over p = 17  Over p = 47  2 3 5 7 11 13 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
3. 2_{−} 2_{+} 2_{−} 0 15 0 1^{2} 1_{+} 17 17 1^{2} 1_{i} 47∗ 47∗  22 22 22 22 22 22 22
V_{4} 40 256 1^{2} 1_{+} 17 17 1^{2} 1_{i} 47∗ 47∗ 22 22 22 22 22 22

Discriminant class: 17

∞ 17 47 a_{1} a_{2} a_{3}  Over p = 17  Over p = 47  2 3 5 7 11 13 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
4. 2_{−} 3_{+} 2_{−}  1 198 -1 1^{4} 3_{+} 17 17 1^{2} 1_{i} 47∗ 47∗ 22 4 4 4 4 22  4
C_{4} 16  9589  1^{2} 1_{i} 47∗ 47∗ 4 4 4 4  22 22 4
5. 2_{−} 1_{+} 2_{−}  -1 10 1 1^{2} 1_{+} 17 17 1^{2} 1_{i} 47∗ 47∗ 22 2 4 2 4 22 22 4
D_{4} 5  1  1^{2} 1_{i} 47∗ 47∗ 4 4 2 4 22  22 2
6. 2_{−} 3_{+} 2_{−}  0 -17 0 1^{4} 3_{+} 17 17 1^{2} 1_{i} 47∗ 47∗  4 2 4 2 22 22 2
D_{4} 32  272  1^{2} 1_{i} 47∗ 47∗ 2 2 4 2 22 22  4