Low Degree Fields Unramified Outside of {∞, 2, 3, 13}

Discriminant class: −78

∞ 2 3 13 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 13  5 7 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} 3_{+} 1_{k} 1_{−} 0 78  1^{2} 3_{+} 2 2 1^{2} 1_{k} 3 3 1^{2} 1_{−} 13∗ 13∗ 2 2 2 2  2  2
C_{2} 4  2  2 2
2. 1_{k} 3_{+} 3_{k} 1_{−} 0 3 -10 1^{2} 3_{+} 2 2 1^{3} 3_{k} 3  1^{2} 1_{−} 13∗ 13∗ 2 2 2 2 3 2 3 2
S_{3} 3 3 2 3 2 2

Discriminant class: −39

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

Discriminant class: −26

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13  5 7 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} 3_{k} 0_{+} 1_{−} 0 26  1^{2} 3_{k} −2∗ −2∗  1^{2} 1_{−} 13∗ 13∗  2  2 2 2
C_{2} 6  2  2 2 2 2
9. 1_{k} 3_{k} 0_{+} 1_{−} 0 -1 -2 1^{2} 3_{k} −2∗ −2∗ 3 0_{+} 1  1^{2} 1_{−} 13∗ 13∗ 3 3 2 3 2 2 2
S_{3}  3 2 3 3 2 2 2 2

Discriminant class: −13

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13 5 7 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} 2_{i} 0_{+} 1_{+} 0 13  1^{2} 2_{i} −∗ −∗ 2 0_{+} ∗ ∗ 1^{2} 1_{+} 13 13 2  2
C_{2} 2  2 2 2
11. 1_{k} 2_{i} 4_{+} 1_{+} 0 9 -36 1^{2} 2_{i} −∗ −∗ 1^{3} 4_{+} ∗  1^{2} 1_{+} 13 13 2 3 3  3 2 3
S_{3}  2 2 2  3 3 3

Discriminant class: −6

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13 5 7 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
12. 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
13. 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 13 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 13 5 7 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} 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
15. 1_{k} 2_{+} 3_{i} 0_{+} 0 0 -2 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  3 0_{+} 1  2 3 2 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∗  3 0_{+} 1  2 3 2 2 3 2 2 3
S_{3}  3 2 3 2 2 2
17. 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
18. 1_{k} 2_{+} 5_{i} 0_{+} 0 0 -6 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 3 0_{+} 1 2 2 2 3 2 2 3
S_{3}  2 3 2 2 2 3 3
19. 1_{k} 2_{+} 1_{i} 2_{+} 0 0 -26 1^{3} 2_{+} ∗ 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1 2 3 2 2 2 2 3
S_{3} 3  2 3 2 2 2 3 3
20. 1_{k} 0_{+} 3_{i} 2_{+} 0 0 -13 2 0_{+} ∗ ∗ 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 2 2 2 3 2 2 3
S_{3} 3  3 2 3 2 2 2 3 3
21. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -52 1^{3} 2_{+} ∗ 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 2 3 2 2 3 2 2 3
S_{3} 3  3 2 3 2 2 2
22. 1_{k} 0_{+} 5_{i} 2_{+} 0 0 -39 2 0_{+} ∗ ∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 3 2 2  2 2
S_{3} 6  3 2  2 2 2 3 3
23. 1_{k} 0_{+} 5_{i} 2_{+} 0 0 -117 2 0_{+} ∗ ∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 3 2 2 3 2 2 3
S_{3} 3  2 3 2 2 2 3 3
24. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -234 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 2 3 2 2 3 2 2 3
S_{3} 6  3 2 3 2 2 2 3 3
25. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -468 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 2 2 2 2 2 3
S_{3} 9  3 2 3 2 2 2
26. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -78 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2  2 2 3 2 2
S_{3} 3  3 2  2 2 2 3 3
27. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -156 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 3 2 2 3 2 2
S_{3} 3  2  2 2 2

Discriminant class: −2

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13 5 7 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
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} 4_{+} 0_{+} 0 -3 -10 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  2 0_{+} ∗ ∗ 2 2 3 3 3 2 2 2
S_{3} 3  2 3 3 2 2 3 2 3

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13 5 7 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
30. 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
31. 1_{k} 2_{i} 4_{+} 0_{+} 0 -3 -4 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗  3 0_{+} 1  3 2 2 3 2 2 3 2
S_{3}  3  2 2  2 3 2
32. 1_{k} 2_{i} 0_{+} 2_{+} -1 -4 12 1^{2} 2_{i} −1 −1 2 0_{+} ∗ ∗ 1^{3} 2_{+} 1  3 2 2 3 2 2 3 2
S_{3} 3  3 3 2 2 3 2 3 2
33. 1_{k} 2_{i} 4_{+} 2_{+} 0 -39 -104 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗  1^{3} 2_{+} 1  3 2 2  2 2  2
S_{3} 6  3 3 2 2 3 2  2
34. 1_{k} 2_{i} 4_{+} 2_{+} 0 -39 -130 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗  1^{3} 2_{+} 1  2 2 3 2 2 3 2
S_{3} 3  3 2 2 3 2 3 2

Discriminant class: 1

∞ 2 3 13 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 13 5 7 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_{+} 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
36. 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
37. 0_{+} 0_{+} 4_{+} 2_{+} 0 -39 91 3 0_{+} 1  1^{3} 4_{+} 1  1^{3} 2_{+} 1  3  3 3 3  3 3
C_{3} 3  3  3  3
38. 0_{+} 0_{+} 4_{+} 2_{+} 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: 2

∞ 2 3 13 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 13 5 7 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_{+} 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 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13 5 7 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
40. 0_{+} 2_{i} 1_{k} 0_{+} 0 -3  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{k} 3 3  2 2  2 2  2 2
C_{2}  2 2  2  2
41. 0_{+} 2_{i} 3_{k} 2_{+} 0 -39 -78 1^{2} 2_{i} −∗ −∗ 1^{3} 3_{k} 3  1^{3} 2_{+} 1  2 2 3 2 2 3 2 2
S_{3} 3  2 2 3 2 3  2

Discriminant class: 6

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13 5 7 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
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: 13

∞ 2 3 13 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 13 5 7 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
44. 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
45. 0_{+} 2_{+} 4_{+} 1_{+} 0 -12 -10 1^{3} 2_{+} ∗  1^{3} 4_{+} 1  1^{2} 1_{+} 13 13 2 2 2 3 2 3  2
S_{3} 3  2 2 3 2 3 2 3 2

Discriminant class: 26

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13  5 7 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
46. 0_{+} 3_{−} 0_{+} 1_{−} 0 -26  1^{2} 3_{−} 2∗ 2∗ 2 0_{+} ∗ ∗ 1^{2} 1_{−} 13∗ 13∗  2  2 2
C_{2} 2  2 2 2 2  2

Discriminant class: 39

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13 5 7 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
47. 0_{+} 2_{i} 1_{k} 1_{+} 0 -39  1^{2} 2_{i} −1 −1 1^{2} 1_{k} 3 3 1^{2} 1_{+} 13 13  2 2  2
C_{2} 2  2  2 2 2 2

Discriminant class: 78

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13  5 7 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
48. 0_{+} 3_{i} 1_{i} 1_{−} 0 -78  1^{2} 3_{i} −2 −2 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{−} 13∗ 13∗ 2  2 2
C_{2} 2  2  2 2
49. 0_{+} 3_{i} 3_{i} 1_{−} 0 -9 -2 1^{2} 3_{i} −2 −2 1^{3} 3_{i} 3∗  1^{2} 1_{−} 13∗ 13∗ 2 3 3 2 2 3 3 3
S_{3}  3  3 2 3 3 2 2