Quartic Fields Ramifying Above {13, 19}

Discriminant class: −247

∞ 13 19 a_{1} a_{2} a_{3}  Over p = 13  Over p = 19  2 3 5 7 11 17 23 29
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
1. 1_{k} 3_{+} 1_{i} -2 -5 19 1^{4} 3_{+} 13∗ 13∗ 1^{2} 1_{i} 19∗ 19∗ 3 4 4 4 2 3 3 2
S_{4} 2 -17 2 0_{+} ∗ ∗ 3 3 3 3 2 4 3 3

Discriminant class: 1

∞ 13 19 a_{1} a_{2} a_{3} Over p = 13  Over p = 19 2 3 5 7 11 17 23 29
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
2. 2_{−} 2_{−} 2_{+} 0 3 0 2^{2} 2_{−} 1 V 2^{2} 2_{+} 1 V 22 22 22 22 22  22
V_{4} 3 64 22 22 22 22 22 22

Discriminant class: 13

∞ 13 19 a_{1} a_{2} a_{3}  Over p = 13  Over p = 19 2 3 5 7 11 17 23 29
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
3. 0_{+} 3_{−} 2_{−} -1 -63 -61 1^{4} 3_{−} 13 13 2^{2} 2_{−} ∗ ∗ 4 22 4 4 4 22 22 22
C_{4}  263  4 4 4 22 4 22 4