Quartic Fields Ramifying Above {11, 31}

Discriminant class: −31

∞ 11 31 a_{1} a_{2} a_{3} Over p = 11  Over p = 31 2 3 5 7 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_{−} 1_{k} -1 -2 -3 2^{2} 2_{−} ∗ ∗ 1^{2} 1_{k} 31 31 3 4 3 3 2 4 3 2
S_{4} -2 2 0_{+} ∗ ∗ 4 4 3 4 22 4 3 2

Discriminant class: −11

∞ 11 31 a_{1} a_{2} a_{3}  Over p = 11  Over p = 31 2 3 5 7 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. 1_{k} 3_{i} 2_{+} -2 18 -17 1^{4} 3_{i} 11∗ 11 2^{2} 2_{+} 1 V 4 22  4 2 2 4 22
D_{4} -13 2 22 4 2 22 22 2
3. 1_{k} 3_{i} 2_{+} -2 -26 -39 1^{4} 3_{i} 11∗ 11 1^{3} 2_{+} 1  2 22 3 4 2 4 2 3
S_{4} 20 4 3 2 2 3 3 3 4

Discriminant class: 1

∞ 11 31 a_{1} a_{2} a_{3} Over p = 11  Over p = 31  2 3 5 7 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
4. 2_{−} 2_{+} 2_{−} 0 21 0 2^{2} 2_{+} 1 V 1^{2} 1_{i} 31∗ 31∗ 22 22  22 22 22 22 22
V_{4} 3 25 1^{2} 1_{i} 31∗ 31∗ 22 22 22 22 22 22

Discriminant class: 341

∞ 11 31 a_{1} a_{2} a_{3}  Over p = 11  Over p = 31 2 3 5 7 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
5. 2_{−} 3_{k} 1_{k} 1 10 -2 1^{4} 3_{k} 11 11∗ 1^{2} 1_{k} 31 31 4 2  4 22 22 4 2
D_{4}  15  2 0_{+} ∗ ∗ 22 2 4 22 22 2  22