Low Degree Fields Unramified Outside of {∞, 2, 3, 7}

Discriminant class: −42

∞ 2 3 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 7 5 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_{k} 1_{k} 1_{k} 0 42  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{k} 3 3 1^{2} 1_{k} 7 7 2 2  2
C_{2} 4  2  2
2. 1_{k} 3_{k} 3_{k} 1_{k} 0 -6 -16 1^{2} 3_{k} −2∗ −2∗ 1^{3} 3_{k} 3  1^{2} 1_{k} 7 7 2 2 3 3 2 3 3 3
S_{3} 2 3 2 3 3 3

Discriminant class: −21

∞ 2 3 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 7 5 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
3. 1_{k} 2_{i} 1_{i} 1_{k} 0 21  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{k} 7 7  2  2
C_{2} 4  2 2 2 2 2 2
4. 1_{k} 2_{i} 3_{i} 1_{k} 0 -6 -12 1^{2} 2_{i} −∗ −∗ 1^{3} 3_{i} 3∗ 1^{2} 1_{k} 7 7 3 3 2 3 3 2 3
S_{3}  3 3 2 2 2 2 2 2
5. 1_{k} 2_{i} 5_{i} 1_{k} 0 9 -30 1^{2} 2_{i} −∗ −∗ 1^{3} 5_{i} 3∗ 1^{2} 1_{k} 7 7 3 2 3 3 3 2 3
S_{3}  3 3 2 2 2 2 2 2
6. 1_{k} 2_{i} 5_{i} 1_{k} 0 18 -12 1^{2} 2_{i} −∗ −∗ 1^{3} 5_{i} 3∗ 1^{2} 1_{k} 7 7 3 3 2 3 3 2
S_{3}  2 2 2 2 2 2
7. 1_{k} 2_{i} 5_{i} 1_{k} 0 9 -12 1^{2} 2_{i} −∗ −∗ 1^{3} 5_{i} 3∗ 1^{2} 1_{k} 7 7 3 2 3 3 2 3
S_{3}  3 3 2 2 2 2 2 2

Discriminant class: −14

∞ 2 3 7 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 7 5 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_{+} 0_{+} 1_{i} 0 14  1^{2} 3_{+} 2 2  1^{2} 1_{i} 7∗ 7∗  2  2  2 2
C_{2} 4  2 2 2 2 2  2
9. 1_{k} 3_{+} 4_{+} 1_{i} 0 -3 -26 1^{2} 3_{+} 2 2 1^{3} 4_{+} 1  1^{2} 1_{i} 7∗ 7∗ 3 2  2 3 3 2 2
S_{3} 3  2 2 2 2 2  3 2

Discriminant class: −7

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

Discriminant class: −6

∞ 2 3 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 7 5 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} 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
13. 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
14. 1_{k} 3_{−} 1_{k} 2_{+} -1 -2 -6 1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 3 3 1^{3} 2_{+} 1  3 3 2 2 2 2 3 3
S_{3} 3  2 2 2 2 3 3 2 2
15. 1_{k} 3_{−} 3_{k} 2_{+} 0 -21 -70 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{k} 3  1^{3} 2_{+} 1  3 2 2 2 2 3 3
S_{3} 3  2 2 2 2  3 2 2
16. 1_{k} 3_{−} 3_{k} 2_{+} 0 -21 -42 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{k} 3  1^{3} 2_{+} 1  3  2 2 2 2 3
S_{3} 3  2 2 2 2 3 3 2 2

Discriminant class: −3

∞ 2 3 7 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 7 5 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
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 2 3 2 3 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 2 3 2 3 2 2 3
S_{3}  3 2 3 2 2 2
20. 1_{k} 2_{+} 1_{i} 2_{+} -1 5  1 1^{3} 2_{+} ∗  1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1  2 2 3 2 3 2 2 3
S_{3} 3  3 2 3 2 2 2  3
21. 1_{k} 2_{+} 5_{i} 0_{+} 0 0 -12 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  3 0_{+} 1  2 2  2  2 2 3
S_{3}  3 2 3 2 2 2 3 3
22. 1_{k} 2_{+} 5_{i} 0_{+} 0 0 -6 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 2 2 3 2 3 2 2 3
S_{3}  2 3 2 2 2 3 3
23. 1_{k} 0_{+} 3_{i} 2_{+} 0 0 -7 2 0_{+} ∗ ∗ 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 2 2 3 2 2 2 3
S_{3} 3  3 2 3 2 2 2 3 3
24. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -14 1^{3} 2_{+} ∗ 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 2 2 2 3 2 2 3
S_{3} 3  2 3 2 2 2 3
25. 1_{k} 0_{+} 5_{i} 2_{+} 0 0 -21 2 0_{+} ∗ ∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2  2 3 2 2 3
S_{3} 3  3 2  2 2 2 3 3
26. 1_{k} 0_{+} 5_{i} 2_{+} 0 0 -63 2 0_{+} ∗ ∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2 3 2 2
S_{3} 6  2 3 2 2 2 3 3
27. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -42 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2 3 2 2 3
S_{3} 3  3 2  2 2 2 3
28. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -252 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2  2 3 2 2
S_{3} 6  3 2 3 2 2 2  3
29. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -84 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2  2 2 3
S_{3} 3  2  2 2 2  3
30. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -126 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2  2 2
S_{3} 9  3 2 3 2 2 2 3

∞ 2 3 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 7 5 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
31. 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
32. 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

Discriminant class: −1

∞ 2 3 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 7 5 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 0_{+} ∗ ∗  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_{+} ∗  2 0_{+} ∗ ∗ 3 2 3 3 2 2 3 2
S_{3}  3  2 2  2 3 2

Discriminant class: 1

∞ 2 3 7 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 7 5 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_{+} 0_{+} 2_{+} -1 -2 1 3 0_{+} 1  3 0_{+} 1  1^{3} 2_{+} 1  3 3  3 3 3  3
C_{3}  3  3 3 3 3 3
36. 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
37. 0_{+} 0_{+} 4_{+} 2_{+} 0 -21 -28  1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3 3 3
C_{3} 3  3 3 3  3
38. 0_{+} 0_{+} 4_{+} 2_{+} 0 -21 35 3 0_{+} 1  1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 3 3  3 3
C_{3} 3  3 3 3 3 3 3 3 3

Discriminant class: 2

∞ 2 3 7 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 7 5 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
39. 0_{+} 3_{+} 0_{+} 0_{+} 0 -2  1^{2} 3_{+} 2 2 2 0_{+} ∗ ∗  2 2 2  2  2
C_{2}  2  2  2 2 2 2
40. 0_{+} 3_{+} 4_{+} 2_{+} 0 -21 -14 1^{2} 3_{+} 2 2 1^{3} 4_{+} ∗  1^{3} 2_{+} 1  2 2 2  2 3 2 3
S_{3} 3  2 3 2 3 2 2 2 2

Discriminant class: 3

∞ 2 3 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 7 5 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
41. 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 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 7 5 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 0_{+} ∗ ∗  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∗  2 0_{+} ∗ ∗ 3 2 2 2 3 3 3 2
S_{3}  2 2  3 3 2 2 3

Discriminant class: 7

∞ 2 3 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 7 5 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_{+} 2_{i} 0_{+} 1_{k} 0 -7  1^{2} 2_{i} −1 −1  1^{2} 1_{k} 7 7 2 2 2 2  2
C_{2}  2 2  2 2

Discriminant class: 14

∞ 2 3 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 7 5 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_{+} 3_{i} 0_{+} 1_{k} 0 -14  1^{2} 3_{i} −2 −2 2 0_{+} ∗ ∗ 1^{2} 1_{k} 7 7  2 2 2 2
C_{2}  2 2  2 2

Discriminant class: 21

∞ 2 3 7 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 7  5 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
46. 0_{+} 0_{+} 1_{k} 1_{i} 0 -21  2 0_{+} ∗ ∗ 1^{2} 1_{k} 3 3 1^{2} 1_{i} 7∗ 7∗  2 2  2 2 2 2
C_{2}  2  2
47. 0_{+} 2_{+} 3_{k} 1_{i} 0 -6 -2 1^{3} 2_{+} ∗  1^{3} 3_{k} 3  1^{2} 1_{i} 7∗ 7∗ 3 2 2 3 2 2 2 2
S_{3}  3 3 3 3 2 3 2

Discriminant class: 42

∞ 2 3 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 7  5 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
48. 0_{+} 3_{−} 1_{i} 1_{i} 0 -42  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{i} 7∗ 7∗ 2  2  2
C_{2} 2  2  2  2  2
49. 0_{+} 3_{−} 5_{i} 1_{i} 0 -27 -30 1^{2} 3_{−} 2∗ 2∗ 1^{3} 5_{i} 3∗  1^{2} 1_{i} 7∗ 7∗ 2 3 3 3  2 3 2
S_{3}  2 3 2 3 3 2 3 2