Low Degree Fields Unramified Outside of {∞, 2, 3, 5}

Discriminant class: −30

∞ 2 3 5 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 5 7 11 13 17 19 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_{+} 1_{i} 1_{+} 0 30  1^{2} 3_{+} 2 2 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 5 5 2  2
C_{2} 4  2  2  2
2. 1_{k} 3_{+} 3_{i} 1_{+} 0 3 -6 1^{2} 3_{+} 2 2 1^{3} 3_{i} 3∗  1^{2} 1_{+} 5 5 2 3 3 3 2 3 3 3
S_{3}  2 3 3 2  2
3. 1_{k} 3_{+} 5_{i} 1_{+} 0 18 -24 1^{2} 3_{+} 2 2 1^{3} 5_{i} 3∗  1^{2} 1_{+} 5 5 2 3 3  2 3  3
S_{3}  3 2  3 2 3 2 3
4. 1_{k} 3_{+} 5_{i} 1_{+} 0 -18 -48 1^{2} 3_{+} 2 2 1^{3} 5_{i} 3∗  1^{2} 1_{+} 5 5 2 3  3 2 3 3
S_{3}  3 2 3  2 3 2 3
5. 1_{k} 3_{+} 5_{i} 1_{+} 0 27 -18 1^{2} 3_{+} 2 2 1^{3} 5_{i} 3∗  1^{2} 1_{+} 5 5 2  3 3 2  3 3
S_{3}  3 2 3 3 2 3 2 3

Discriminant class: −15

∞ 2 3 5 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 5 7 11 13 17 19 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_{+} 1_{k} 1_{−} 0 15  1^{2} 1_{k} 3 3 1^{2} 1_{−} 5∗ 5∗ 2 2 2  2
C_{2} 2  2 2 2  2  2
7. 1_{k} 0_{+} 3_{k} 1_{−} 0 3 -1 3 0_{+} 1  1^{3} 3_{k} 3  1^{2} 1_{−} 5∗ 5∗ 2 2 2 3 3 3 2 3
S_{3}  2 2 2  3 2 3 2

Discriminant class: −10

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  7 11 13 17 19 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_{k} 0_{+} 1_{−} 0 10  1^{2} 3_{k} −2∗ −2∗ 2 0_{+} ∗ ∗ 1^{2} 1_{−} 5∗ 5∗  2  2 2
C_{2} 2  2  2 2
9. 1_{k} 3_{k} 4_{+} 1_{−} 0 -3 -22 1^{2} 3_{k} −2∗ −2∗ 1^{3} 4_{+} ∗  1^{2} 1_{−} 5∗ 5∗ 3  3 2 3 3 2 2
S_{3}  3 2 3 3 3 2 2

Discriminant class: −6

∞ 2 3 5 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 5 7 11 13 17 19 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} 3_{−} 1_{k} 0_{+} 0 6  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 3 3  2 2 2 2
C_{2} 2  2 2 2 2  2 2
11. 1_{k} 3_{−} 3_{k} 0_{+} 0 3 -2 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{k} 3  3 0_{+} 1  3 3 2 2 2 2  3
S_{3}  2 2 2 2 3  2 2

Discriminant class: −5

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 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
12. 1_{k} 2_{i} 0_{+} 1_{+} 0 5  1^{2} 2_{i} −∗ −∗  1^{2} 1_{+} 5 5  2 2 2 2  2
C_{2} 2  2  2 2
13. 1_{k} 2_{i} 4_{+} 1_{+} 0 -3 -8 1^{2} 2_{i} −∗ −∗ 1^{3} 4_{+} 1  1^{2} 1_{+} 5 5 3 2 2 2 2 3 3 2
S_{3} 3  2 3  3 2 2 3 3

Discriminant class: −3

∞ 2 3 5 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 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} 0_{+} 1_{i} 0_{+} 1 1  2 0_{+} ∗ ∗ 1^{2} 1_{i} 3∗ 3∗ 2 0_{+} ∗ ∗  2  2  2 2
C_{2}  2  2 2 2
15. 1_{k} 2_{+} 3_{i} 0_{+} 0 0 -2 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  2 0_{+} ∗ ∗ 3 2 3 2 3 2 2
S_{3}  3 2  2 2 2 3 3
16. 1_{k} 0_{+} 5_{i} 0_{+} 0 0 -3 2 0_{+} ∗ ∗ 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 3 2 3 2 3 2 2 3
S_{3}  3 2 3 2 2 2
17. 1_{k} 2_{+} 1_{i} 2_{+} -1 -3 -3 1^{3} 2_{+} ∗ 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗ 3 2 3 2 3 2 2 3
S_{3}  2 3 2 2 2 3 3
18. 1_{k} 0_{+} 3_{i} 2_{+} 0 0 -5 2 0_{+} ∗ ∗ 1^{3} 3_{i} 3∗ 1^{3} 2_{+} ∗ 3 2 2 3 2 2 3
S_{3}  3 2 3 2 2 2 3
19. 1_{k} 2_{+} 5_{i} 0_{+} 0 0 -6 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 2 0_{+} ∗ ∗ 2 3 2 3 2 2 3
S_{3}  2 3 2 2 2 3 3
20. 1_{k} 2_{+} 5_{i} 0_{+} 0 0 -12 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 2 0_{+} ∗ ∗ 3 2 2 2 2 3
S_{3}  3 2 3 2 2 2 3 3
21. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -20 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  2 3 2  2 2 3
S_{3} 3  3 2 3 2 2 2  3
22. 1_{k} 0_{+} 5_{i} 2_{+} 0 0 -15 2 0_{+} ∗ ∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 3 2 3 2 2
S_{3} 2  3 2 3 2 2 2 3
23. 1_{k} 0_{+} 5_{i} 2_{+} 0 0 -45 2 0_{+} ∗ ∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  3 2 3 2  2 2 3
S_{3}  2  2 2 2 3
24. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -90 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2  2 3 2 2 3
S_{3} 3  3 2  2 2 2 3 3
25. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -150 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  3 2 3 2 3 2 2 3
S_{3} 3  3 2  2 2 2  3
26. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -30 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  3 2 3 2  2 2
S_{3} 3  3 2 3 2 2 2 3 3
27. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -60 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  3 2  2 3 2 2
S_{3} 3  2 3 2 2 2  3

Discriminant class: −2

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 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
28. 1_{k} 3_{i} 0_{+} 0_{+} 0 2  1^{2} 3_{i} −2 −2  2 0_{+} ∗ ∗ 2  2  2 2 2
C_{2}  2  2 2  2
29. 1_{k} 3_{i} 0_{+} 2_{+} -1 2 2 1^{2} 3_{i} −2 −2 3 0_{+} 1  1^{3} 2_{+} ∗  2 3 2 3 3 2 2 2
S_{3}  2 3  2 2  2 3

Discriminant class: −2 (continued)

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 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
30. 1_{k} 3_{i} 4_{+} 0_{+} 0 -3 -10 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  2 0_{+} ∗ ∗ 2 3 2 3 3 2 2 2
S_{3} 3  2 3 3 2 2 3 2 3
31. 1_{k} 3_{i} 4_{+} 2_{+} 0 -30 -80 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1 1^{3} 2_{+} ∗ 2 2 3 2 2 2
S_{3} 3  2 3 3 2 2 3 2 3
32. 1_{k} 3_{i} 4_{+} 2_{+} 0 15 -10 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  2 3 2 3  2 2 2
S_{3} 3  2  3 2 2 3 2

Discriminant class: −1

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 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
33. 1_{k} 2_{i} 0_{+} 0_{+} 0 1  1^{2} 2_{i} −1 −1 2 0_{+} ∗ ∗  2 2  2 2  2
C_{2}  2 2  2  2
34. 1_{k} 2_{i} 4_{+} 0_{+} 0 -3 -4 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗  3 0_{+} 1  2 2 3 3 2 2 3 2
S_{3}  3  2 2  2 3 2

Discriminant class: 1

∞ 2 3 5 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 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_{+} 0_{+} 4_{+} 0_{+} 0 -3 1 3 0_{+} 1  1^{3} 4_{+} 1  3 0_{+} 1  3 3 3  3 3 3
C_{3}  3 3 3  3 3 3

Discriminant class: 2

∞ 2 3 5 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 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. 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
37. 0_{+} 3_{+} 4_{+} 2_{+} 0 -30 -40 1^{2} 3_{+} 2 2 1^{3} 4_{+} ∗  1^{3} 2_{+} ∗  3 2 2 3 2  2
S_{3}  2 3 2 3 2 2 2 2

Discriminant class: 3

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 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_{+} 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
39. 0_{+} 2_{i} 3_{k} 2_{+} 0 -15 -20 1^{2} 2_{i} −∗ −∗ 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  2 3 3 2 2 3 2 2
S_{3}  3 2 2 3 2 3 3 2

Discriminant class: 5

∞ 2 3 5 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 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_{+} 0_{+} 0_{+} 1_{+} 1 -1  2 0_{+} ∗ ∗ 2 0_{+} ∗ ∗ 1^{2} 1_{+} 5 5 2  2 2  2
C_{2}  2  2 2 2  2
41. 0_{+} 2_{+} 4_{+} 1_{+} 0 -12 -14 1^{3} 2_{+} ∗  1^{3} 4_{+} ∗  1^{2} 1_{+} 5 5 2 3 2 2 3 2 3 3
S_{3}  2 3 2 2 2 3  2

Discriminant class: 6

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 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_{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
43. 0_{+} 3_{k} 5_{i} 0_{+} 0 -9 -6 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗  3 0_{+} 1  2 2 2 2 3 3 3 2
S_{3}  2 2  3 3 2 2 3

Discriminant class: 10

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  7 11 13 17 19 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_{−} 0_{+} 1_{−} 0 -10  1^{2} 3_{−} 2∗ 2∗  1^{2} 1_{−} 5∗ 5∗ 2 2  2 2 2 2
C_{2} 2  2  2 2

Discriminant class: 15

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  7 11 13 17 19 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
45. 0_{+} 2_{i} 1_{i} 1_{−} 0 -15  1^{2} 2_{i} −1 −1 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{−} 5∗ 5∗  2  2 2 2 2
C_{2} 2  2 2  2
46. 0_{+} 2_{i} 5_{i} 1_{−} 0 -18 -12 1^{2} 2_{i} −1 −1 1^{3} 5_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 3 3 2 3 2 2 2 2
S_{3}  2 2 3 2 3  3 3

Discriminant class: 30

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 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
47. 0_{+} 3_{i} 1_{k} 1_{+} 0 -30  1^{2} 3_{i} −2 −2 1^{2} 1_{k} 3 3 1^{2} 1_{+} 5 5  2  2  2
C_{2} 2  2 2 2 2 2 2 2