Low Degree Fields Ramifying Above {3, 7, 19}

Discriminant class: −399

∞ 3 7 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 19 2 5 11 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_{i} 1_{i} 1_{k} 0 399  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{k} 19 19  2  2
C_{2} 16  2 2  2 2 2
2. 1_{k} 3_{i} 1_{i} 1_{k} 0 9 -5 1^{3} 3_{i} 3∗  1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{k} 19 19 3  2 3 3 2 3
S_{3}  2 2 3 3  2 2 2
3. 1_{k} 5_{i} 1_{i} 1_{k} 0 -27 -117 1^{3} 5_{i} 3∗  1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{k} 19 19 3 3 2  3 2 3 3
S_{3}  2 2 3 3 3 2 2 2
4. 1_{k} 5_{i} 1_{i} 1_{k} 0 9 -138 1^{3} 5_{i} 3∗  1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{k} 19 19  3 2 3  2 3 3
S_{3}  2 2 3  3 2 2 2
5. 1_{k} 5_{i} 1_{i} 1_{k} 0 9 -33 1^{3} 5_{i} 3∗  1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{k} 19 19 3 3 2 3 3 2  3
S_{3}  2 2  3 3 2 2 2

Discriminant class: −19

∞ 3 7 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 7  Over p = 19  2 5 11 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
6. 1_{k} 4_{+} 2_{+} 1_{i} 0 -21 -112 1^{3} 4_{+} ∗  1^{3} 2_{+} 1  1^{2} 1_{i} 19∗ 19∗ 2 3  2 3  2 2
S_{3} 3  2 2 3 3 2 2  2
7. 1_{k} 4_{+} 2_{+} 1_{i} 0 63 -252 1^{3} 4_{+} ∗  1^{3} 2_{+} 1  1^{2} 1_{i} 19∗ 19∗ 2 3 3 2 3 3 2 2
S_{3} 3  2 2  2 2 3 2

Discriminant class: −3

∞ 3 7 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 19 2 5 11 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
8. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -133 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2 3 2 2 2 3
S_{3} 9  3 2  2 2 2
9. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -931 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2  2 2 2
S_{3} 18  2 3 2 2 2 3 3
10. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -7581 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} 1 2 2 2 3 2 2 2 3
S_{3} 9  3 2 3 2 2 2 3 3
11. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -2793 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2 3 2 2 2 3
S_{3} 36  3 2  2 2 2 3 3
12. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1197 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2  2 2 2 3
S_{3} 9  3 2 3 2 2 2
13. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -399 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} 1 2 2 2 3 2 2 2
S_{3} 9  2 3 2 2 2

Discriminant class: 1

∞ 3 7 19 a_{1} a_{2}  a_{3} Over p = 3  Over p = 7  Over p = 19 2 5 11 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
14. 0_{+} 4_{+} 2_{+} 2_{+} 0 -399 -3059 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3
C_{3} 9  3  3 3  3 3 3
15. 0_{+} 4_{+} 2_{+} 2_{+} 0 -399 665 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3  3  3 3
C_{3} 9  3 3  3 3 3 3 3
16. 0_{+} 4_{+} 2_{+} 2_{+} 0 -399 -2926 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3  3  3  3
C_{3} 36  3  3 3 3 3 3
17. 0_{+} 4_{+} 2_{+} 2_{+} 0 -399 1729 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3  3 3 3 3
C_{3} 9  3 3  3

Discriminant class: 57

∞ 3 7 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 7  Over p = 19  2 5 11 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
18. 0_{+} 3_{k} 2_{+} 1_{i} 0 -21 -21 1^{3} 3_{k} 3  1^{3} 2_{+} 1  1^{2} 1_{i} 19∗ 19∗ 3 2 2 2 2 2 3 2
S_{3} 3  2  3 2 3  3 2