Low Degree Fields Ramifying Above {2, 3, 19}

Discriminant class: −114

∞ 2 3 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 19  5 7 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} 3_{i} 1_{k} 1_{i} 0 114  1^{2} 3_{i} −2 −2 1^{2} 1_{k} 3 3 1^{2} 1_{i} 19∗ 19∗  2 2  2  2
C_{2} 8  2  2 2
2. 1_{k} 3_{i} 3_{k} 1_{i} 0 6 -24 1^{2} 3_{i} −2 −2 1^{3} 3_{k} 3  1^{2} 1_{i} 19∗ 19∗ 3 2 2 3 2  2
S_{3}  3 3 3 2 3 2 2

Discriminant class: −57

∞ 2 3 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 19 5 7 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
3. 1_{k} 2_{i} 1_{i} 1_{k} 0 57  1^{2} 2_{i} −1 −1 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{k} 19 19 2 2  2 2
C_{2} 4  2  2  2
4. 1_{k} 2_{i} 3_{i} 1_{k} 0 9 -14 1^{2} 2_{i} −1 −1 1^{3} 3_{i} 3∗  1^{2} 1_{k} 19 19 2 2 3 2 2  3 3
S_{3}  2 3 2 3 3 2 3 3
5. 1_{k} 2_{i} 5_{i} 1_{k} 0 9 -24 1^{2} 2_{i} −1 −1 1^{3} 5_{i} 3∗  1^{2} 1_{k} 19 19 2 2 3 2 2 3 3
S_{3} 2  2  2 3 3 2 3 3
6. 1_{k} 2_{i} 5_{i} 1_{k} 0 -18 -60 1^{2} 2_{i} −1 −1 1^{3} 5_{i} 3∗  1^{2} 1_{k} 19 19 2 2  2 2 3  3
S_{3}  2 3 2 3 3 2 3 3
7. 1_{k} 2_{i} 5_{i} 1_{k} 0 -27 -60 1^{2} 2_{i} −1 −1 1^{3} 5_{i} 3∗  1^{2} 1_{k} 19 19 2 2 3 2 2 3 3 3
S_{3} 2 3 2 2

Discriminant class: −19

∞ 2 3 19 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 19  5 7 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} 2_{+} 4_{+} 1_{i} 0 6 -14 1^{3} 2_{+} ∗  1^{3} 4_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 3  3 2  3 2 2
S_{3} 3  2 2 3  2 2 3 2
9. 1_{k} 2_{+} 4_{+} 1_{i} 0 -12 -22 1^{3} 2_{+} ∗  1^{3} 4_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 3 3  2 3 3 2 2
S_{3} 3  2 2  3 2 2  2

Discriminant class: −3

∞ 2 3 19 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 19 5 7 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
10. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -76 1^{3} 2_{+} ∗ 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 2 2 3 2 2 2 3
S_{3} 6  3 2 3 2 2 2 3
11. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -38 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  2 3 2  2 2 2 3
S_{3} 3  2 3 2 2 2  3
12. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -228 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 3 2 3 2 2 2 3
S_{3} 9  2 3 2 2 2 3
13. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -114 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 3 2 3 2 2 2 3
S_{3} 3  3 2 3 2 2 2  3

Discriminant class: −3 (continued)

∞ 2 3 19 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 19 5 7 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. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -684 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 3 2  2 2 2
S_{3} 6  3 2  2 2 2 3
15. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -342 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2  2 3 2 2 2
S_{3} 27  3 2  2 2 2  3

Discriminant class: −2

∞ 2 3 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 19 5 7 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
16. 1_{k} 3_{i} 4_{+} 2_{+} 0 -57 -494 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  1^{3} 2_{+} 1  2 2  2 3 2 2 2
S_{3} 9  2 3 3 2 2 3 2 3
17. 1_{k} 3_{i} 4_{+} 2_{+} 0 -57 -190 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  1^{3} 2_{+} 1  2 2 3 2 3 2 2 2
S_{3} 9  2  3 2 2  2

Discriminant class: 6

∞ 2 3 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 19 5 7 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} 3_{i} 2_{+} 0 -57 38 1^{2} 3_{k} −2∗ −2∗ 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  3 2 2 2 2 3 3 2
S_{3} 3  2 2 3  2 2 3
19. 0_{+} 3_{k} 5_{i} 2_{+} 0 -171 -798 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  3 2 2 2 2 3  2
S_{3} 3  2 2 3 3 3 2 2
20. 0_{+} 3_{k} 5_{i} 2_{+} 0 -171 -570 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 2 2 2 2 3 2
S_{3} 3  2 2 3 3 3 2 2 3

Discriminant class: 114

∞ 2 3 19 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 19 5 7 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
21. 0_{+} 3_{+} 1_{i} 1_{k} 0 -114 1^{2} 3_{+} 2 2 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{k} 19 19 2 2 2 2
C_{2} 2  2 2 2 2 2
22. 0_{+} 3_{+} 3_{i} 1_{k} 0 -18 -16 1^{2} 3_{+} 2 2 1^{3} 3_{i} 3∗ 1^{2} 1_{k} 19 19 3 3 3 2 2 2 2
S_{3}  3 3 2 2 2 2 2