Quartic Fields Ramifying Above {7, 47}

Discriminant class: −47

∞ 7 47 a_{1} a_{2} a_{3}  Over p = 7  Over p = 47  2 3 5 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. 1_{k} 2_{−} 3_{k}  0 47 0 1^{2} 1_{i} 7∗ 7∗ 1^{4} 3_{k} 47∗ 47 22 22 2 4 2 22 2 4
D_{4} 3 -188 1^{2} 1_{i} 7∗ 7∗ 4 2 22 2 4 22 22

Discriminant class: −7

∞ 7 47 a_{1} a_{2} a_{3}  Over p = 7  Over p = 47 2 3 5 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. 1_{k} 1_{i} 2_{+} 0 -1 -3 1^{2} 1_{i} 7∗ 7∗ 1^{3} 2_{+} ∗  3 4 4 3 2 4 4 3
S_{4} -4 22 2 3 4 3 3 2 4

Discriminant class: 1

∞ 7 47 a_{1} a_{2} a_{3}  Over p = 7  Over p = 47 2 3 5 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
3. 2_{−} 2_{−} 2_{+} 0 27 0 1^{2} 1_{i} 7∗ 7∗ 2^{2} 2_{+} 1 V  22 22 22 22 22 22 22
V_{4} 5 100 1^{2} 1_{i} 7∗ 7∗ 22 22 22 22 22 22

Discriminant class: 329

∞ 7 47 a_{1} a_{2} a_{3}  Over p = 7  Over p = 47  2 3 5 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
4. 2_{−} 1_{i} 3_{i} -2 25 -24 1^{2} 1_{i} 7∗ 7∗ 1^{4} 3_{i} 47 47∗ 22 2 22 4 22 2 22 4
D_{4} 15  191  4 22 22 22 4  2 2