Low Degree Fields Ramifying Above {3, 19}

Discriminant class: −19

∞ 3 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 19  2 5 7 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} 4_{+} 1_{i} 0 6 -5 1^{3} 4_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 2  3 3 2 3 3 2
S_{3}  2 2 2 3 3 2 2 3

Discriminant class: −3

∞ 3 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 19 2 5 7 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. 1_{k} 1_{i} 2_{+} -1 -6 -12 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1  2 2 3 2 3 2 2 2
S_{3} 3  3 3 2 3 2 2 2 3
3. 1_{k} 5_{i} 2_{+} 0 0 -57 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 2 2 2 2 2 2
S_{3} 6  3 3 2 3 2 2 2 3
4. 1_{k} 5_{i} 2_{+} 0 0 -171 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2 3 2 2 2
S_{3} 6  2  2 2 2 3

Discriminant class: 1

∞ 3 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 19 2 5 7 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
5. 0_{+} 4_{+} 2_{+} 0 -57 152 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 3 3  3  3
C_{3} 3  3  3  3 3 3 3
6. 0_{+} 4_{+} 2_{+} 0 -57 19 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3  3 3 3 3 3
C_{3} 3  3  3  3

Discriminant class: 57

∞ 3 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 19  2 5 7 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
7. 0_{+} 1_{k} 1_{i} 0 -57  1^{2} 1_{k} 3 3 1^{2} 1_{i} 19∗ 19∗  2  2 2 2 2
C_{2}  2 2  2