Low Degree Fields Unramified Outside of {∞, 3, 13, 23}

Discriminant class: −299

∞ 3 13 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 23  2 5 7 11 17 19 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_{+} 1_{i} 0 299  1^{2} 1_{+} 13 13 1^{2} 1_{i} 23∗ 23∗ 2  2  2
C_{2} 8  2 2 2 2 2 2
2. 1_{k} 4_{+} 1_{+} 1_{i} 0 -9 -180 1^{3} 4_{+} 1 1^{2} 1_{+} 13 13 1^{2} 1_{i} 23∗ 23∗ 2 3 3 2 3 2
S_{3} 3  3 2 2 2 2 2 2 3

Discriminant class: −39

∞ 3 13 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 13 Over p = 23 2 5 7 11 17 19 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} 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
4. 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
5. 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
6. 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
7. 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
8. 1_{k} 1_{i} 1_{+} 2_{+} -1  8 -29 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2 3 2 2 2 2
S_{3}  2 3 3  2 3 3 2
9. 1_{k} 3_{i} 1_{+} 2_{+} 0 69 -115 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 3 2 3 3 2
10. 1_{k} 3_{i} 1_{+} 2_{+} 0 -69 -276 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  2
11. 1_{k} 5_{i} 1_{+} 2_{+} 0 207 -483 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 2 3  2
12. 1_{k} 5_{i} 1_{+} 2_{+} 0 207 -2208 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  2 3 2 2 2 2
S_{3} 3  2 3 3  2 3 3 2
13. 1_{k} 5_{i} 1_{+} 2_{+} 0 207 -1311 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 3 2 3  2
14. 1_{k} 5_{i} 1_{+} 2_{+} 0 -207 -1518 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 2  2 2 2 2
S_{3} 6  2  3 3 2 3 3 2
15. 1_{k} 5_{i} 1_{+} 2_{+} 0 -207 -1173 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2 3 2 2 2 2
S_{3} 6  2 3  3 2  3 2
16. 1_{k} 5_{i} 1_{+} 2_{+} 0 -207 -2967 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2  2 2 2 2
S_{3} 3  2  2  3 2

Discriminant class: −23

∞ 3 13 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 23  2 5 7 11 17 19 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_{+} 0_{+} 1_{i} 0 23  1^{2} 1_{i} 23∗ 23∗  2 2 2 2 2
C_{2} 3  2  2  2  2 2
18. 1_{k} 0_{+} 0_{+} 1_{i} -1  0  1 3 0_{+} 1  3 0_{+} 1  1^{2} 1_{i} 23∗ 23∗ 3 2 2 2 2 2 3 3
S_{3}  2 3 2 3 2  2 2
19. 1_{k} 4_{+} 2_{+} 1_{i} 0 -39 -442 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{2} 1_{i} 23∗ 23∗  2 2 2 2 2 3
S_{3} 9  2  2 3 2 3 2 2
20. 1_{k} 4_{+} 2_{+} 1_{i} 0 -117 -585 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{2} 1_{i} 23∗ 23∗ 3 2 2 2 2 2  3
S_{3} 9  2 3 2 3 2 3 2 2
21. 1_{k} 4_{+} 2_{+} 1_{i} 0 -39 -143 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{2} 1_{i} 23∗ 23∗ 3 2 2 2 2 2 3 3
S_{3} 9  2 3 2  2 3 2 2

Discriminant class: −3

∞ 3 13 23 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 13 Over p = 23 2 5 7 11 17 19 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
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  -23 1^{3} 3_{i} 3∗  3 0_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 2 3 2
S_{3}  2 3 2 2 2  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 -207 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 2 3 2 2 3 2 3
S_{3} 8  3 2  2 2 2  3
29. 1_{k} 5_{i} 0_{+} 2_{+} 0 0  -69 1^{3} 5_{i} 3∗  3 0_{+} 1  1^{3} 2_{+} ∗  2 2  2 2  2 3
S_{3}  3 2 3 2 2 2  3
30. 1_{k} 1_{i} 2_{+} 2_{+} -1 100 -144 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 2  2 3
S_{3} 3  3 2 3 2 2 2 3
31. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -299 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 2 3 2 3
S_{3} 6  3 2  2 2 2 3 3
32. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -897 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 2 3 2
S_{3} 18  3 2 3 2 2 2 3 3
33. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -2691 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 3 2 2 2 3
S_{3} 6  2 3 2 2 2 3 3
34. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -20631 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 3
35. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -11661 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 2 3 2 3
S_{3} 3  2  2 2 2 3

Discriminant class: 1

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

Discriminant class: 13

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

Discriminant class: 69

∞ 3 13 23 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 13  Over p = 23 2 5 7 11 17 19 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_{+} 1_{i} 0_{+} 1_{k} 0 -69  1^{2} 1_{i} 3∗ 3∗  1^{2} 1_{k} 23 23 2  2  2 2
C_{2}  2 2 2 2  2 2 2
42. 0_{+} 3_{i} 0_{+} 1_{k} 0 -6  -3 1^{3} 3_{i} 3∗  3 0_{+} 1  1^{2} 1_{k} 23 23 2 3 2 3 3 2 2 3
S_{3}  2 2 2 2 3 2 2 2
43. 0_{+} 5_{i} 2_{+} 1_{k} 0 -117 -312 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{2} 1_{k} 23 23 2 3 2 3 3 2 2 3
S_{3} 3  2 2 2 2  2 2 2
44. 0_{+} 5_{i} 2_{+} 1_{k} 0 -234 -1365 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{2} 1_{k} 23 23 2  2 3  2 2 3
S_{3} 3  2 2 2 2 3 2 2 2
45. 0_{+} 5_{i} 2_{+} 1_{k} 0 -234 -429 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{2} 1_{k} 23 23 2 3 2 3 2 2
S_{3} 3  2 2 2 2 3 2 2 2

Discriminant class: 897

∞ 3 13 23 a_{1} a_{2} a_{3}  Over p = 3  Over p = 13  Over p = 23 2 5 7 11 17 19 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_{+} 1_{i} 1_{+} 1_{k} 0 -897  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 13 13 1^{2} 1_{k} 23 23  2  2  2 2
C_{2} 4  2  2
47. 0_{+} 5_{i} 1_{+} 1_{k} 0 -27 -15 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{2} 1_{k} 23 23 3 2 3 2 3 3 2 2
S_{3}  3  2 3 3  2 3