Quartic Fields Ramifying Above {11, 13}

Discriminant class: 1

∞ 11 13 a_{1} a_{2} a_{3} Over p = 11  Over p = 13 2 3 5 7 17 19 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. 2_{−} 2_{+} 2_{−} 0 -1 0 2^{2} 2_{+} 1 V 2^{2} 2_{−} 1 V 22  22 22 22 22  22
V_{4} 5  36  22 22 22 22 22  22 22

Discriminant class: 13

∞ 11 13 a_{1} a_{2} a_{3} Over p = 11  Over p = 13 2 3 5 7 17 19 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. 0_{+} 2_{−} 3_{−} -1 -37 -35 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{−} 13 13 4  4 4  4 22 22
C_{4}  81  4 4 4  4  4 22
3. 2_{−} 2_{+} 3_{−} -1 2 -22 1^{3} 2_{+} ∗  1^{4} 3_{−} 13 13 4 3 2 2 3 4 3 3
S_{4} 8  29  2 4 2 3 4 3 2 3