Low Degree Fields Unramified Outside of {∞, 3, 13, 29}

Discriminant class: −1131

∞ 3 13 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 29  2 5 7 11 17 19 23 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_{k} 1_{+} 1_{−} 0 1131  1^{2} 1_{k} 3 3 1^{2} 1_{+} 13 13 1^{2} 1_{−} 29∗ 29∗ 2  2 2  2
C_{2} 8  2 2 2 2  2 2
2. 1_{k} 3_{k} 1_{+} 1_{−} 0 12 -11 1^{3} 3_{k} 3  1^{2} 1_{+} 13 13 1^{2} 1_{−} 29∗ 29∗ 2 3 2 2 3 3 2 3
S_{3} 2 2 2 2 2 2

Discriminant class: −87

∞ 3 13 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 29  2 5 7 11 17 19 23 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_{k} 0_{+} 1_{−} 0 87  1^{2} 1_{k} 3 3  1^{2} 1_{−} 29∗ 29∗  2  2 2 2
C_{2} 6  2  2  2 2 2
4. 1_{k} 1_{k} 0_{+} 1_{−} -1 2 1 1^{2} 1_{k} 3 3 3 0_{+} 1  1^{2} 1_{−} 29∗ 29∗ 3 2 3 3 3 2 2 2
S_{3}  2  2 3 2 2 2 3
5. 1_{k} 1_{k} 2_{+} 1_{−} -1 22 51 1^{2} 1_{k} 3 3 1^{3} 2_{+} 1  1^{2} 1_{−} 29∗ 29∗ 3 2 3  2 2 2
S_{3} 3  2 3 2  2 2 2 3
6. 1_{k} 1_{k} 2_{+} 1_{−} -1 -4 25 1^{2} 1_{k} 3 3 1^{3} 2_{+} 1  1^{2} 1_{−} 29∗ 29∗ 3 2  3 3 2 2 2
S_{3} 6  2 3 2 3 2 2 2 3
7. 1_{k} 1_{k} 2_{+} 1_{−} -1 -4 -92 1^{2} 1_{k} 3 3 1^{3} 2_{+} 1 1^{2} 1_{−} 29∗ 29∗ 2 3 3 3 2 2 2
S_{3} 3  2 3 2 3 2 2 2

Discriminant class: −39

∞ 3 13 29 a_{1} a_{2} a_{3} Over p = 3 Over p = 13 Over p = 29 2 5 7 11 17 19 23 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} 1_{i} 1_{+} 0_{+} 0 39  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 13 13 2 0_{+} ∗ ∗  2  2 2 2 2
C_{2} 4  2  2  2
9. 1_{k} 3_{i} 1_{+} 0_{+} 0  3  -3 1^{3} 3_{i} 3∗  1^{2} 1_{+} 13 13 2 0_{+} ∗ ∗ 3 3 2  2 2 2 2
S_{3}  2  3 3 2 3 3 2
10. 1_{k} 5_{i} 1_{+} 0_{+} 0 -9 -15 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 2 0_{+} ∗ ∗ 3  2 3 2 2 2 2
S_{3}  2 3  3 2  3 2
11. 1_{k} 5_{i} 1_{+} 0_{+} 0 -9 -24 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 2 0_{+} ∗ ∗  3 2 3 2 2 2 2
S_{3}  2 3 3 3 2 3 3 2
12. 1_{k} 5_{i} 1_{+} 0_{+} 0  9  -3 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 2 0_{+} ∗ ∗ 3 3 2 3 2 2 2 2
S_{3}  2 3 3  2 3  2
13. 1_{k} 1_{i} 1_{+} 2_{+} -1 10 -72 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 2 3 2 2 2 2
S_{3}  2 3  3 2 3 3 2
14. 1_{k} 3_{i} 1_{+} 2_{+} 0 87 -29 1^{3} 3_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2 3 2 2 2 2
S_{3} 6  2 3 3 3 2  3 2
15. 1_{k} 3_{i} 1_{+} 2_{+} 0 -87 -609 1^{3} 3_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3  2 3 2 2 2 2
S_{3} 3  2 3 3  2 3  2
16. 1_{k} 5_{i} 1_{+} 2_{+} 0 -261 -1740 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  2 3 2 2 2 2
S_{3} 21  2  3 3 2 3 3 2

Discriminant class: −39 (continued)

∞ 3 13 29 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 13  Over p = 29 2 5 7 11 17 19 23 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} 5_{i} 1_{+} 2_{+} 0 261 -3045 1^{3} 5_{i} 3∗ 1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗ 3 2 2 2 2 2
S_{3} 3  2 3 3 3 2 3 3 2
18. 1_{k} 5_{i} 1_{+} 2_{+} 0 261 -3741 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2 3 2 2 2 2
S_{3} 3  2  2 3  2
19. 1_{k} 5_{i} 1_{+} 2_{+} 0 -261 -1653 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2  2 2 2 2
S_{3} 6  2 3  3 2 3 3 2
20. 1_{k} 5_{i} 1_{+} 2_{+} 0 261 -1914 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 2  2 2 2 2
S_{3} 15  2 3 3  2  2
21. 1_{k} 5_{i} 1_{+} 2_{+} 0 261 -1479 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2 3 2 2 2 2
S_{3} 3  2  3 3 2  3 2

Discriminant class: −3

∞ 3 13 29 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 13 Over p = 29 2 5 7 11 17 19 23 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
22. 1_{k} 1_{i} 0_{+} 0_{+} 1 1  1^{2} 1_{i} 3∗ 3∗  2 0_{+} ∗ ∗ 2 2  2 2  2
C_{2}  2  2 2 2
23. 1_{k} 5_{i} 0_{+} 0_{+} 0 0  -3 1^{3} 5_{i} 3∗  3 0_{+} 1  2 0_{+} ∗ ∗ 2 2 3 2 2 3 2 3
S_{3}  3 2 3 2 2 2
24. 1_{k} 3_{i} 2_{+} 0_{+} 0 0 -13 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 2 0_{+} ∗ ∗ 2 2 2 2 3 2 3
S_{3} 3  3 2 3 2 2 2 3 3
25. 1_{k} 3_{i} 0_{+} 2_{+} 0 0 -29 1^{3} 3_{i} 3∗ 3 0_{+} 1 1^{3} 2_{+} ∗ 2 2 2 2 3 2
S_{3}  2 3 2 2 2 3 3
26. 1_{k} 5_{i} 2_{+} 0_{+} 0 0  -39 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 0_{+} ∗ ∗ 2 2 3 2 2  2
S_{3} 6  3 2  2 2 2 3 3
27. 1_{k} 5_{i} 2_{+} 0_{+} 0 0 -117 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 0_{+} ∗ ∗ 2 2 3 2 2 3 2 3
S_{3} 3  2 3 2 2 2 3 3
28. 1_{k} 5_{i} 0_{+} 2_{+} 0 0 -261 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 2 2 3 2 2 3 2 3
S_{3}  3 2 3 2 2 2 3 3
29. 1_{k} 5_{i} 0_{+} 2_{+} 0 0  -87 1^{3} 5_{i} 3∗  3 0_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 2  2 3
S_{3}  3 2  2 2 2 3 3
30. 1_{k} 1_{i} 2_{+} 2_{+} 0 0 -377 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 2 3 2 3
S_{3} 3  3 2 3 2 2 2
31. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -4901 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 2  2 3
S_{3} 24  3 2  2 2 2 3 3
32. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -3393 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 2  2 3
S_{3} 3  2  2 2 2
33. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -32799 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 2 3 2
S_{3} 15  3 2 3 2 2 2 3 3
34. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -14703 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 3 2 2 3 2 3
S_{3} 18  2 3 2 2 2 3 3
35. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1131 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 3 2 2 3 2
S_{3} 12  3 2 3 2 2 2

Discriminant class: 1

∞ 3 13 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 13 Over p = 29 2 5 7 11 17 19 23 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_{+} 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
37. 0_{+} 0_{+} 2_{+} 0_{+} -1 -4 -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
38. 0_{+} 4_{+} 2_{+} 0_{+} 0 -39 91 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 0_{+} 1  3 3  3 3 3  3
C_{3} 3  3  3  3
39. 0_{+} 4_{+} 2_{+} 0_{+} 0 -39 -26 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 3  3 3 3 3
C_{3} 3  3 3 3 3  3 3

Discriminant class: 13

∞ 3 13 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 29 2 5 7 11 17 19 23 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_{+} 1_{+} 0_{+} 1 -3  1^{2} 1_{+} 13 13  2 2 2 2  2  2
C_{2}  2 2  2  2  2

Discriminant class: 29

∞ 3 13 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 29 2 5 7 11 17 19 23 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_{+} 0_{+} 0_{+} 1_{+} 0 -29  2 0_{+} ∗ ∗  1^{2} 1_{+} 29 29 2  2 2 2  2
C_{2}  2 2 2 2  2
42. 0_{+} 4_{+} 0_{+} 1_{+} 0 -12 -13 1^{3} 4_{+} ∗  1^{2} 1_{+} 29 29 2 3 3 2 2 2 3 2
S_{3}  2 2 2 2 3 3 2 3

Discriminant class: 377

∞ 3 13 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 29 2 5 7 11 17 19 23 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_{+} 1_{+} 0 -377  2 0_{+} ∗ ∗ 1^{2} 1_{+} 13 13 1^{2} 1_{+} 29 29  2 2  2
C_{2} 2  2  2 2 2