Low Degree Fields Unramified Outside of {∞, 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 -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
6. 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
7. 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

Discriminant class: −38

∞ 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} 3_{−} 0_{+} 1_{k} 0 38  1^{2} 3_{−} 2∗ 2∗  1^{2} 1_{k} 19 19 2  2  2
C_{2} 6  2 2  2
9. 1_{k} 3_{−} 0_{+} 1_{k} -1 -2 -2 1^{2} 3_{−} 2∗ 2∗ 3 0_{+} 1  1^{2} 1_{k} 19 19 2 3 2 3 3 3 3 2
S_{3} 2 2 3 3 2 3

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
10. 1_{k} 0_{+} 0_{+} 1_{i} -1 5  2 0_{+} ∗ ∗ 2 0_{+} ∗ ∗ 1^{2} 1_{i} 19∗ 19∗  2  2 2
C_{2}  2 2  2 2  2
11. 1_{k} 2_{+} 0_{+} 1_{i} 0 -2 -2 1^{3} 2_{+} ∗  2 0_{+} ∗ ∗ 1^{2} 1_{i} 19∗ 19∗ 3 3 3 2 3  2 2
S_{3}  2 2 3 3 2 2 3 2
12. 1_{k} 0_{+} 4_{+} 1_{i} 0 6 -5 2 0_{+} ∗ ∗ 1^{3} 4_{+} ∗ 1^{2} 1_{i} 19∗ 19∗ 3 3 2 3 3 2 2
S_{3}  2 2 3 3 2 2 3 2
13. 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: −19 (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_{+} 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

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
15. 1_{k} 3_{−} 1_{k} 0_{+} 0 6  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 3 3 2 0_{+} ∗ ∗  2 2 2
C_{2} 2  2 2 2 2  2 2
16. 1_{k} 3_{−} 3_{k} 0_{+} 0 3 -2 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{k} 3  2 0_{+} ∗ ∗ 3 3 3 2 2 2  3
S_{3}  2 2 2 2 3  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
17. 1_{k} 0_{+} 1_{i} 0_{+} 1 1  2 0_{+} ∗ ∗ 1^{2} 1_{i} 3∗ 3∗  2  2  2 2 2
C_{2}  2  2 2 2
18. 1_{k} 2_{+} 3_{i} 0_{+} 0 0 -2 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  3 0_{+} 1  2 3 2 3 2 2 2
S_{3}  3 2  2 2 2 3 3
19. 1_{k} 0_{+} 5_{i} 0_{+} 0 0 -3 2 0_{+} ∗ ∗ 1^{3} 5_{i} 3∗  3 0_{+} 1  2 3 2 3 2 2 2 3
S_{3}  3 2 3 2 2 2
20. 1_{k} 2_{+} 5_{i} 0_{+} 0 0 -6 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 3 0_{+} 1 2 2 3 2 2 2 3
S_{3}  2 3 2 2 2 3 3
21. 1_{k} 2_{+} 5_{i} 0_{+} 0 0 -12 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 2 3 2 2 2 2 3
S_{3}  3 2 3 2 2 2 3 3
22. 1_{k} 0_{+} 1_{i} 2_{+} -1 -6 -12 2 0_{+} ∗ ∗ 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1  2 3 2 3 2 2 2 3
S_{3} 3  3 2 3 2 2 2 3 3
23. 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
24. 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
25. 1_{k} 0_{+} 5_{i} 2_{+} 0 0 -171 2 0_{+} ∗ ∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 3 2 3 2 2 2
S_{3} 6  2  2 2 2 3 3
26. 1_{k} 0_{+} 5_{i} 2_{+} 0 0 -57 2 0_{+} ∗ ∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2  2  2 2 2 3
S_{3} 6  3 2 3 2 2 2 3 3
27. 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
28. 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
29. 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
30. 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

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
31. 1_{k} 3_{i} 0_{+} 0_{+} 0 2  1^{2} 3_{i} −2 −2  2 2  2  2 2 2
C_{2}  2  2 2  2
32. 1_{k} 3_{i} 4_{+} 0_{+} 0 -3 -10 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  3 0_{+} 1  2 2 3 2 3 2 2 2
S_{3} 3  2 3 3 2 2 3 2 3
33. 1_{k} 3_{i} 0_{+} 2_{+} -1 13  7 1^{2} 3_{i} −2 −2 3 0_{+} 1  1^{3} 2_{+} 1  2 2 3 2  2 2 2
S_{3} 3  2 3  2 2 3 2 3
34. 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
35. 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

Discriminant class: −1

∞ 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
36. 1_{k} 2_{i} 0_{+} 0_{+} 0 1  1^{2} 2_{i} −1 −1 2 0_{+} ∗ ∗ 2 0_{+} ∗ ∗  2 2  2  2
C_{2}  2 2  2  2
37. 1_{k} 2_{i} 4_{+} 0_{+} 0 -3 -4 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗  2 0_{+} ∗ ∗ 3 2 2 3 3 2 3 2
S_{3}  3  2 2  2 3 2

Discriminant class: 1

∞ 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
38. 0_{+} 0_{+} 4_{+} 0_{+} 0 -3 1 3 0_{+} 1  1^{3} 4_{+} 1  3 3 3 3  3 3 3
C_{3}  3 3 3  3 3 3
39. 0_{+} 0_{+} 0_{+} 2_{+} -1 -6 7 3 0_{+} 1  3 0_{+} 1  1^{3} 2_{+} 1  3  3 3 3 3
C_{3}  3 3 3 3 3 3 3
40. 0_{+} 0_{+} 4_{+} 2_{+} 0 -57 19 3 0_{+} 1  1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3  3
C_{3} 3  3  3  3
41. 0_{+} 0_{+} 4_{+} 2_{+} 0 -57 152  1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 3 3  3  3 3
C_{3} 3  3  3 3 3 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
42. 0_{+} 3_{+} 0_{+} 0_{+} 0 -2  1^{2} 3_{+} 2 2 2 0_{+} ∗ ∗ 2 0_{+} ∗ ∗ 2  2 2  2
C_{2}  2  2  2 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
43. 0_{+} 2_{i} 1_{k} 0_{+} 0 -3  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{k} 3 3 2 0_{+} ∗ ∗ 2 2  2  2 2
C_{2}  2 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
44. 0_{+} 3_{k} 1_{i} 0_{+} 0 -6  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{i} 3∗ 3∗  2 2 2 2  2
C_{2}  2 2  2 2
45. 0_{+} 3_{k} 5_{i} 0_{+} 0 -9 -6 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗  3 0_{+} 1  3 2 2 2 2 3 3 2
S_{3}  2 2  3 3 2 2 3
46. 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
47. 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
48. 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

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
49. 0_{+} 2_{i} 0_{+} 1_{k} 0 -19  1^{2} 2_{i} −∗ −∗  1^{2} 1_{k} 19 19  2 2 2  2 2
C_{2}  2 2 2 2 2

Discriminant class: 38

∞ 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
50. 0_{+} 3_{k} 0_{+} 1_{i} 0 -38  1^{2} 3_{k} −2∗ −2∗ 2 0_{+} ∗ ∗ 1^{2} 1_{i} 19∗ 19∗ 2 2  2
C_{2}  2  2  2 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
51. 0_{+} 0_{+} 1_{k} 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

∞ 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
52. 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
53. 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