Low Degree Fields Unramified Outside of {∞, 3, 13, 19}

Discriminant class: −247

∞ 3 13 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 19 2 5 7 11 17 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_{−} 1_{k} 0 247  2 0_{+} ∗ ∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19  2 2 2  2
C_{2} 6  2 2
2. 1_{k} 0_{+} 1_{−} 1_{k} 0 1 -3 2 0_{+} ∗ ∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19 3 2 2 2 3 3 2 3
S_{3}  3 3 3 2 2 3 3 3

Discriminant class: −39

∞ 3 13 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 13 Over p = 19 2 5 7 11 17 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} 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

Discriminant class: −19

∞ 3 13 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 19  2 5 7 11 17 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_{+} 0_{+} 1_{i} -1 5  2 0_{+} ∗ ∗ 2 0_{+} ∗ ∗ 1^{2} 1_{i} 19∗ 19∗ 2  2 2
C_{2}  2 2  2 2  2
9. 1_{k} 4_{+} 0_{+} 1_{i} 0 6 -5 1^{3} 4_{+} ∗  2 0_{+} ∗ ∗ 1^{2} 1_{i} 19∗ 19∗ 2  3 3 3 3 2 2
S_{3}  2 2 3 3 2 2 3 2

Discriminant class: −3

∞ 3 13 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 13 Over p = 19 2 5 7 11 17 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 3 2 2 2 2 3
S_{3}  3 2 3 2 2 2
12. 1_{k} 1_{i} 0_{+} 2_{+} -1 -6 -12 1^{2} 1_{i} 3∗ 3∗ 3 0_{+} 1 1^{3} 2_{+} 1 2 2 3 2 2 2 2 3
S_{3} 3  3 2 3 2 2 2 3 3
13. 1_{k} 3_{i} 2_{+} 0_{+} 0 0 -13 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

Discriminant class: −3 (continued)

∞ 3 13 19 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 13 Over p = 19 2 5 7 11 17 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  -39 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2 2 2 2
S_{3} 6  3 2  2 2 2 3 3
15. 1_{k} 5_{i} 2_{+} 0_{+} 0 0 -117 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  3 0_{+} 1  2 2 3 2 2 2 2 3
S_{3} 3  2 3 2 2 2 3 3
16. 1_{k} 5_{i} 0_{+} 2_{+} 0 0 -171 1^{3} 5_{i} 3∗  3 0_{+} 1  1^{3} 2_{+} 1  2 2 3 2 2 2 2
S_{3} 6  2  2 2 2 3 3
17. 1_{k} 5_{i} 0_{+} 2_{+} 0 0  -57 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2  2 2 2 2 3
S_{3} 6  3 2 3 2 2 2 3 3
18. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -247 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 3 2 2 2 2
S_{3} 9  3 2  2 2 2  3
19. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -3211 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 3 2 2 2 2 3
S_{3} 9  2 3 2 2 2 3
20. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -14079 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} 1 2 2 3 2 2 2 2 3
S_{3} 18  3 2 3 2 2 2 3
21. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -9633 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2  2 2 2 2
S_{3} 18  3 2  2 2 2 3
22. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -2223 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 3 2 2 2 2 3
S_{3} 9  3 2 3 2 2 2  3
23. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -741 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2  2 2 2 2 3
S_{3} 72  2 3 2 2 2  3

Discriminant class: 1

∞ 3 13 19 a_{1} a_{2}  a_{3} Over p = 3  Over p = 13 Over p = 19 2 5 7 11 17 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_{+} 4_{+} 0_{+} 0_{+} 0 -3  1 1^{3} 4_{+} 1  3 0_{+} 1  3 3 3 3  3 3 3
C_{3}  3 3 3  3 3 3
25. 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
26. 0_{+} 0_{+} 0_{+} 2_{+} -1 -6  7 3 0_{+} 1  3 0_{+} 1  1^{3} 2_{+} 1  3 3  3 3 3
C_{3}  3 3 3 3 3 3 3
27. 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
28. 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
29. 0_{+} 4_{+} 0_{+} 2_{+} 0 -57  19 1^{3} 4_{+} 1  3 0_{+} 1  1^{3} 2_{+} 1  3  3 3 3 3  3
C_{3} 3  3  3  3
30. 0_{+} 4_{+} 0_{+} 2_{+} 0 -57 152 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 3 3 3  3 3
C_{3} 3  3  3 3 3 3
31. 0_{+} 0_{+} 2_{+} 2_{+} -1 -82 311 3 0_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3 3
C_{3} 3  3 3  3 3  3
32. 0_{+} 0_{+} 2_{+} 2_{+} -1 -82  64  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3 3
C_{3} 3  3  3 3 3 3  3
33. 0_{+} 4_{+} 2_{+} 2_{+} 0 -741 4940 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
34. 0_{+} 4_{+} 2_{+} 2_{+} 0 -741 7163 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3  3 3 3  3
C_{3} 9  3 3 3 3 3  3 3

Discriminant class: 1 (continued)

∞ 3 13 19 a_{1} a_{2}  a_{3} Over p = 3  Over p = 13 Over p = 19 2 5 7 11 17 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 -741 6175 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
36. 0_{+} 4_{+} 2_{+} 2_{+} 0 -741 -2717 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3  3 3 3
C_{3} 9  3  3 3 3 3  3

Discriminant class: 13

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

Discriminant class: 57

∞ 3 13 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 19  2 5 7 11 17 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} 0_{+} 1_{i} 0 -57  1^{2} 1_{k} 3 3 2 0_{+} ∗ ∗ 1^{2} 1_{i} 19∗ 19∗  2  2 2 2  2
C_{2}  2  2  2

Discriminant class: 741

∞ 3 13 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 19 2 5 7 11 17 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_{−} 1_{k} 0 -741  1^{2} 1_{k} 3 3 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19 2  2  2 2
C_{2} 2  2  2
40. 0_{+} 3_{k} 1_{−} 1_{k} 0 -12 -3 1^{3} 3_{k} 3  1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19 2 3 2 3 2 2 3
S_{3}  3 2 3  3 2 3 3