Quartic Fields Ramifying Above {7, 31}

Discriminant class: −31

∞ 7 31 a_{1} a_{2} a_{3}  Over p = 7  Over p = 31  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 37 41 43 47 53 59 61
1. 1_{k} 2_{−} 3_{k}  -2 17 -16 1^{2} 1_{i} 7∗ 7∗ 1^{4} 3_{k} 31∗ 31 22 2 22 4 2 2 22 4
D_{4} -153 1^{2} 1_{i} 7∗ 7∗ 4 4 22 4 22 4 22 2

Discriminant class: 1

∞ 7 31 a_{1} a_{2} a_{3}  Over p = 7  Over p = 31 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 37 41 43 47 53 59 61
2. 2_{−} 2_{−} 2_{+} 0 19 0 1^{2} 1_{i} 7∗ 7∗ 2^{2} 2_{+} 1 V  22 22 22 22 22 22 22
V_{4} 3 36 1^{2} 1_{i} 7∗ 7∗ 22 22 22 22 22 22 22 22

Discriminant class: 217

∞ 7 31 a_{1} a_{2} a_{3}  Over p = 7  Over p = 31  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 37 41 43 47 53 59 61
3. 2_{−} 1_{i} 3_{i} 1 12 -29 1^{2} 1_{i} 7∗ 7∗ 1^{4} 3_{i} 31 31∗ 22 22 2 4 22 22 2 4
D_{4} 3  35  4 4 2 4 2 4 2 22