Low Degree Fields Unramified Outside of {∞, 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 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
4. 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
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} 0_{+} 0_{+} 1_{i} -1 5  2 0_{+} ∗ ∗  1^{2} 1_{i} 19∗ 19∗ 2  2  2 2
C_{2}  2 2  2 2  2
7. 1_{k} 0_{+} 2_{+} 1_{i} -1 5 -6 2 0_{+} ∗ ∗ 1^{3} 2_{+} 1  1^{2} 1_{i} 19∗ 19∗ 2 3 3 2  3 2 2
S_{3} 3  2 2 3 3 2 2 3 2
8. 1_{k} 4_{+} 0_{+} 1_{i} 0 6 -5 1^{3} 4_{+} ∗ 3 0_{+} 1 1^{2} 1_{i} 19∗ 19∗ 2 3 2 3 3 2 2
S_{3}  2 2 3 3 2 2 3 2
9. 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
10. 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: −7

∞ 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
11. 1_{k} 0_{+} 1_{i} 0_{+} 1 2  2 0_{+} ∗ ∗ 1^{2} 1_{i} 7∗ 7∗ 2 0_{+} ∗ ∗  2  2 2  2
C_{2}  2  2  2 2
12. 1_{k} 4_{+} 1_{i} 0_{+} 0 -3 -5 1^{3} 4_{+} ∗  1^{2} 1_{i} 7∗ 7∗ 2 0_{+} ∗ ∗ 3 2 3 2 2  2
S_{3}  3 2 3 2 3 2 2 3

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
13. 1_{k} 1_{i} 0_{+} 0_{+} 1 1  1^{2} 1_{i} 3∗ 3∗  2 2 2  2 2 2
C_{2}  2  2 2 2

Discriminant class: −3 (continued)

∞ 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. 1_{k} 5_{i} 0_{+} 0_{+} 0 0 -3 1^{3} 5_{i} 3∗ 3 0_{+} 1 3 0_{+} 1 2 2 2 3 2 2 2 3
S_{3}  3 2 3 2 2 2
15. 1_{k} 1_{i} 0_{+} 2_{+} -1 -6 -12 1^{2} 1_{i} 3∗ 3∗ 3 0_{+} 1 1^{3} 2_{+} 1 2 2 2 3 2 2 2 3
S_{3} 3  3 2 3 2 2 2 3 3
16. 1_{k} 3_{i} 2_{+} 0_{+} 0 0 -7 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 2 2 2 3 2 2 2 3
S_{3} 3  3 2 3 2 2 2 3 3
17. 1_{k} 5_{i} 2_{+} 0_{+} 0 0 -63 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 3 0_{+} 1 2 2 2 3 2 2 2
S_{3} 6  2 3 2 2 2 3 3
18. 1_{k} 5_{i} 2_{+} 0_{+} 0 0 -21 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  3 0_{+} 1  2 2 2  2 2 2 3
S_{3} 3  3 2  2 2 2 3 3
19. 1_{k} 5_{i} 0_{+} 2_{+} 0 0 -171 1^{3} 5_{i} 3∗  3 0_{+} 1  1^{3} 2_{+} 1  2 2 2 3 2 2 2
S_{3} 6  2  2 2 2 3 3
20. 1_{k} 5_{i} 0_{+} 2_{+} 0 0 -57 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 2  2 2 2 3
S_{3} 6  3 2 3 2 2 2 3 3
21. 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
22. 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
23. 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
24. 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
25. 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
26. 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

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
27. 0_{+} 0_{+} 2_{+} 0_{+} -1 -2  1 3 0_{+} 1  1^{3} 2_{+} 1  3 0_{+} 1  3 3 3  3 3  3
C_{3}  3  3 3 3 3 3
28. 0_{+} 4_{+} 0_{+} 0_{+} 0 -3  1 1^{3} 4_{+} 1  3 0_{+} 1  3 3 3 3  3 3 3
C_{3}  3 3 3  3 3 3
29. 0_{+} 0_{+} 0_{+} 2_{+} -1 -6  7 3 0_{+} 1  1^{3} 2_{+} 1  3 3  3 3 3 3
C_{3}  3 3 3 3 3 3 3
30. 0_{+} 4_{+} 2_{+} 0_{+} 0 -21 -28 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 0_{+} 1  3 3 3 3 3 3
C_{3} 3  3 3 3  3
31. 0_{+} 4_{+} 2_{+} 0_{+} 0 -21  35 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 0_{+} 1  3  3 3  3 3
C_{3} 3  3 3 3 3 3 3 3 3
32. 0_{+} 0_{+} 2_{+} 2_{+} -1 -44 -69  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3 3  3 3
C_{3} 3  3 3 3  3  3
33. 0_{+} 0_{+} 2_{+} 2_{+} 1 -44 -64 3 0_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3 3
C_{3} 3  3 3 3 3  3 3
34. 0_{+} 4_{+} 0_{+} 2_{+} 0 -57  19 1^{3} 4_{+} 1  3 0_{+} 1  1^{3} 2_{+} 1  3  3 3 3 3  3
C_{3} 3  3  3  3

Discriminant class: 1 (continued)

∞ 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
35. 0_{+} 4_{+} 0_{+} 2_{+} 0 -57 152 1^{3} 4_{+} 1  3 0_{+} 1  1^{3} 2_{+} 1  3 3  3  3 3
C_{3} 3  3  3 3 3 3
36. 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
37. 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
38. 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
39. 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

Discriminant class: 21

∞ 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
40. 0_{+} 1_{k} 1_{i} 0_{+} 0 -21  1^{2} 1_{k} 3 3 1^{2} 1_{i} 7∗ 7∗ 2 0_{+} ∗ ∗ 2  2 2  2 2 2
C_{2}  2  2

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
41. 0_{+} 1_{k} 0_{+} 1_{i} 0 -57  1^{2} 1_{k} 3 3  1^{2} 1_{i} 19∗ 19∗  2 2 2 2 2  2
C_{2}  2  2  2
42. 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

Discriminant class: 133

∞ 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
43. 0_{+} 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 2