Low Degree Fields Ramifying Above {5, 13, 19}

Discriminant class: −1235

∞ 5 13 19 a_{1} a_{2} a_{3} Over p = 5 Over p = 13 Over p = 19 2 3 7 11 17 23 29 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
1. 1_{k} 1_{−} 1_{+} 1_{k} 0 1235  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{+} 13 13 1^{2} 1_{k} 19 19 2  2 2 2 2
C_{2} 12  2  2  2
2. 1_{k} 1_{−} 1_{+} 1_{k} -1  -5 10 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{+} 13 13 1^{2} 1_{k} 19 19 2 3 3 2 2 2 2 3
S_{3} 2 3 2 3 3 2

Discriminant class: −95

∞ 5 13 19 a_{1} a_{2} a_{3} Over p = 5  Over p = 13  Over p = 19  2 3 7 11 17 23 29 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d  s 37 41 43 47 53 59 61 67
3. 1_{k} 1_{+} 2_{+} 1_{i} 0 -13 -52 1^{2} 1_{+} 5 5 1^{3} 2_{+} 1  1^{2} 1_{i} 19∗ 19∗  3 2 3 2 2 2 2
S_{3} 3  3 2 2 2 3 2 3 3