Low Degree Fields Ramifying Above {13, 31}

Discriminant class: −403

∞ 13 31 a_{1} a_{2} a_{3}  Over p = 13  Over p = 31 2 3 5 7 11 17 19 23
G h  f^{e} c_{w} d  s f^{e} c_{w} d s 29 37 41 43 47 53 59 61
1. 1_{k} 1_{−} 1_{k} 0 403  1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 31 31 2 2 2 2  2 2 2
C_{2} 2  2  2 2 2 2 2 2

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  f^{e} c_{w} d s f^{e} c_{w} d s 29 37 41 43 47 53 59 61
2. 0_{+} 2_{+} 2_{+} 1 -134 209 1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3  3
C_{3} 3  3 3  3  3
3. 0_{+} 2_{+} 2_{+} 1 -134 -597 1^{3} 2_{+} 1  1^{3} 2_{+} 1  3  3 3 3 3 3 3
C_{3} 3  3  3 3  3 3 3