Low Degree Fields Ramifying Above {7, 19}

Discriminant class: −19

∞ 7 19 a_{1} a_{2} a_{3} Over p = 7  Over p = 19  2 3 5 11 13 17 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
1. 1_{k} 2_{+} 1_{i} -1 5 -6 1^{3} 2_{+} 1  1^{2} 1_{i} 19∗ 19∗ 2 2 3 3 2  3 2
S_{3} 3  2 2 2 3 3 2 2 3

Discriminant class: 1

∞ 7 19 a_{1} a_{2} a_{3} Over p = 7  Over p = 19 2 3 5 11 13 17 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
2. 0_{+} 2_{+} 2_{+} 1 -44 -64 1^{3} 2_{+} 1  1^{3} 2_{+} 1  3  3 3 3 3 3
C_{3} 3  3 3 3 3 3  3 3
3. 0_{+} 2_{+} 2_{+} -1 -44 -69 1^{3} 2_{+} 1  1^{3} 2_{+} 1  3  3 3 3  3
C_{3} 3  3 3 3 3  3

Discriminant class: 133

∞ 7 19 a_{1} a_{2} a_{3} Over p = 7 Over p = 19 2 3 5 11 13 17 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
4. 0_{+} 1_{i} 1_{k} 0 -133  1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{k} 19 19 2  2  2  2
C_{2}  2  2 2  2