Low Degree Fields Unramified Outside of {∞, 3, 7, 13}

Discriminant class: −91

∞ 3 7 13 a_{1} a_{2} a_{3} Over p = 3  Over p = 7  Over p = 13  2 5 11 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} 0_{+} 1_{k} 1_{−} 0 91  2 0_{+} ∗ ∗ 1^{2} 1_{k} 7 7 1^{2} 1_{−} 13∗ 13∗ 2  2 2
C_{2} 2  2  2 2
2. 1_{k} 4_{+} 1_{k} 1_{−} 0 -12 -23 1^{3} 4_{+} ∗  1^{2} 1_{k} 7 7 1^{2} 1_{−} 13∗ 13∗ 2 3 2 2 3  3 3
S_{3}  2 3  3  2 2

Discriminant class: −39

∞ 3 7 13 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 13 2 5 11 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} 1_{i} 0_{+} 1_{+} 0 39  1^{2} 1_{i} 3∗ 3∗ 2 0_{+} ∗ ∗ 1^{2} 1_{+} 13 13  2 2 2 2 2
C_{2} 4  2  2  2
4. 1_{k} 3_{i} 0_{+} 1_{+} 0 3 -3 1^{3} 3_{i} 3∗  2 0_{+} ∗ ∗ 1^{2} 1_{+} 13 13 3 3  2 2 2 2 2
S_{3}  2  3 3 2 3 3 2
5. 1_{k} 5_{i} 0_{+} 1_{+} 0 -9 -24 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 1^{2} 1_{+} 13 13  3 3 2 2 2 2 2
S_{3}  2 3 3 3 2 3 3 2
6. 1_{k} 5_{i} 0_{+} 1_{+} 0 -9 -15 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 1^{2} 1_{+} 13 13 3  3 2 2 2 2 2
S_{3}  2 3  3 2  3 2
7. 1_{k} 5_{i} 0_{+} 1_{+} 0 9 -3 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 1^{2} 1_{+} 13 13 3 3 3 2 2 2 2 2
S_{3} 2 3 3 2 3 2

Discriminant class: −7

∞ 3 7 13 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 13 2 5 11 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} 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
9. 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 13 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 13 2 5 11 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} 1_{i} 0_{+} 0_{+} 1 1  1^{2} 1_{i} 3∗ 3∗  2 2 2 2  2 2
C_{2}  2  2 2 2
11. 1_{k} 5_{i} 0_{+} 0_{+} 0 0  -3 1^{3} 5_{i} 3∗  3 0_{+} 1  3 0_{+} 1  2 2 2 2 3 2 2 3
S_{3}  3 2 3 2 2 2
12. 1_{k} 3_{i} 2_{+} 0_{+} 0 0 -7 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 3 0_{+} 1 2 2 2 2 2 2 3
S_{3} 3  3 2 3 2 2 2 3 3
13. 1_{k} 3_{i} 0_{+} 2_{+} 0 0 -13 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 2 2 2 2 3 2 2 3
S_{3} 3  3 2 3 2 2 2 3 3

Discriminant class: −3 (continued)

∞ 3 7 13 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 7  Over p = 13 2 5 11 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} 5_{i} 2_{+} 0_{+} 0 0 -63 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 3 0_{+} 1 2 2 2 2 3 2 2
S_{3} 6  2 3 2 2 2 3 3
15. 1_{k} 5_{i} 2_{+} 0_{+} 0 0 -21 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 2 2 3 2 2 3
S_{3} 3  3 2  2 2 2 3 3
16. 1_{k} 1_{i} 2_{+} 2_{+} 0 0 -91 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2 2 3 2 2
S_{3} 9  3 2 3 2 2 2 3
17. 1_{k} 5_{i} 0_{+} 2_{+} 0 0 -39 1^{3} 5_{i} 3∗  3 0_{+} 1  1^{3} 2_{+} 1  2 2 2 2  2 2
S_{3} 6  3 2  2 2 2 3 3
18. 1_{k} 5_{i} 0_{+} 2_{+} 0 0 -117 1^{3} 5_{i} 3∗  3 0_{+} 1  1^{3} 2_{+} 1  2 2 2 2 3 2 2 3
S_{3} 3  2 3 2 2 2 3 3
19. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -637 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2 2 3 2 2 3
S_{3} 9  2  2 2 2  3
20. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -3549 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2 2 3 2 2
S_{3} 9  3 2 3 2 2 2  3
21. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -819 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2 2 3 2 2 3
S_{3} 9  3 2  2 2 2 3
22. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -273 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2 2  2 2 3
S_{3} 27  2 3 2 2 2 3
23. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1911 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2 2  2 2 3
S_{3} 45  3 2 3 2 2 2  3

Discriminant class: 1

∞ 3 7 13 a_{1} a_{2}  a_{3} Over p = 3  Over p = 7  Over p = 13 2 5 11 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
24. 0_{+} 0_{+} 2_{+} 0_{+} -1 -2  1 3 0_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3 3  3
C_{3}  3  3 3 3 3 3
25. 0_{+} 4_{+} 0_{+} 0_{+} 0 -3  1 1^{3} 4_{+} 1  3 0_{+} 1  3 0_{+} 1  3 3 3  3 3 3
C_{3}  3 3 3  3 3 3
26. 0_{+} 0_{+} 0_{+} 2_{+} -1 -4  -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
27. 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
28. 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
29. 0_{+} 0_{+} 2_{+} 2_{+} -1 -30  64 3 0_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3  3  3 3
C_{3} 3  3 3 3 3  3 3
30. 0_{+} 0_{+} 2_{+} 2_{+} 1 -30  27  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3  3  3 3 3
C_{3} 3  3 3 3 3 3 3
31. 0_{+} 4_{+} 0_{+} 2_{+} 0 -39  91 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3  3 3
C_{3} 3  3  3  3
32. 0_{+} 4_{+} 0_{+} 2_{+} 0 -39 -26 1^{3} 4_{+} 1  3 0_{+} 1  1^{3} 2_{+} 1  3  3 3 3  3
C_{3} 3  3 3 3 3  3 3
33. 0_{+} 4_{+} 2_{+} 2_{+} 0 -273 1547 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
34. 0_{+} 4_{+} 2_{+} 2_{+} 0 -273 -1729 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

Discriminant class: 1 (continued)

∞ 3 7 13 a_{1} a_{2}  a_{3} Over p = 3  Over p = 7  Over p = 13 2 5 11 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_{+} 4_{+} 2_{+} 2_{+} 0 -273 -91 1^{3} 4_{+} 1 1^{3} 2_{+} 1 1^{3} 2_{+} 1 3 3 3 3 3
C_{3} 9  3 3 3 3
36. 0_{+} 4_{+} 2_{+} 2_{+} 0 -273 728 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

Discriminant class: 13

∞ 3 7 13 a_{1} a_{2} a_{3} Over p = 3  Over p = 7  Over p = 13 2 5 11 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
37. 0_{+} 0_{+} 0_{+} 1_{+} 1 -3  2 0_{+} ∗ ∗ 1^{2} 1_{+} 13 13 2 2 2  2  2
C_{2}  2 2  2  2  2

Discriminant class: 21

∞ 3 7 13 a_{1} a_{2} a_{3} Over p = 3  Over p = 7  Over p = 13 2 5 11 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_{+} 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: 273

∞ 3 7 13 a_{1} a_{2} a_{3} Over p = 3  Over p = 7  Over p = 13  2 5 11 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_{+} 1_{k} 1_{k} 1_{−} 0 -273  1^{2} 1_{k} 3 3 1^{2} 1_{k} 7 7 1^{2} 1_{−} 13∗ 13∗  2  2 2
C_{2} 2  2 2  2 2 2 2 2