Low Degree Fields Ramifying Above {3, 11, 19}

Discriminant class: −627

∞ 3 11 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 11 Over p = 19 2 5 7 13 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_{k} 1_{k} 1_{k} 0 627  1^{2} 1_{k} 3 3 1^{2} 1_{k} 11 11 1^{2} 1_{k} 19 19 2 2 2  2 2 2
C_{2} 4  2 2 2 2  2 2
2. 1_{k} 3_{k} 1_{k} 1_{k} 0 6 -43 1^{3} 3_{k} 3  1^{2} 1_{k} 11 11 1^{2} 1_{k} 19 19 2 2 2 3 3 2 2 2
S_{3} 2 2 2 2 3 2 2

Discriminant class: −3

∞ 3 11 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 11 Over p = 19 2 5 7 13 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} 3_{i} 2_{+} 2_{+} 0 0 -2299 1^{3} 3_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 2 3 3 2 2 2 3
S_{3} 9  3 2 3 2 2 2 3 3
4. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -209 1^{3} 3_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 2 2 2 2
S_{3} 27  3 2 3 2 2 2 3
5. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -11913 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2  2 2 2 3
S_{3} 3  2  2 2 2 3 3
6. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -6897 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 3 3 2 2 2
S_{3} 3  3 2 3 2 2 2 3 3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1881 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 3 3 2 2 2 3
S_{3} 6  2  2 2 2 3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -627 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 3 3 2 2 2 3
S_{3} 12  3 2 3 2 2 2 3

Discriminant class: 57

∞ 3 11 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 11  Over p = 19  2 5 7 13 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
9. 0_{+} 3_{k} 2_{+} 1_{i} 0 -33 -55 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 3 2 3 2 2 2  2
S_{3}  2  3 2 3 3 3 2