Quartic Fields Ramifying Above {13, 31}

Discriminant class: −31

∞ 13 31 a_{1} a_{2} a_{3}  Over p = 13  Over p = 31  2 3 5 7 11 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} -1 -19 60 1^{2} 1_{+} 13 13 1^{4} 3_{k} 31∗ 31 3 2 3 3 4 4 3 4
S_{4} 2 -27 1^{2} 1_{−} 13∗ 13∗ 2 2 3 2 22 4 3 4

Discriminant class: 1

∞ 13 31 a_{1} a_{2} a_{3} Over p = 13  Over p = 31 2 3 5 7 11 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 9 0 2^{2} 2_{−} 1 V 2^{2} 2_{+} 1 V 22 22 22 22 22 22 22 22
V_{4} 3 121 22 22 22 22 22 22 22 22
3. 2_{−} 2_{+} 2_{−} -1 -6 7 1^{3} 2_{+} 1  1^{2} 1_{k} 31 31 3 3 22 3 3 3 3 3
A_{4} 4 18 1^{2} 1_{k} 31 31 3 3 3 3 22 3 3

Discriminant class: 13

∞ 13 31 a_{1} a_{2} a_{3}  Over p = 13  Over p = 31 2 3 5 7 11 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. 0_{+} 3_{−} 2_{−} -1 -102 -100 1^{4} 3_{−} 13 13 2^{2} 2_{−} ∗ ∗ 4 22 4 4 4  4
C_{4}  731  22 4 4  4 22 4 22