Low Degree Fields Unramified Outside of {∞, 3, 19, 29}

Discriminant class: −551

∞ 3 19 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 19  Over p = 29  2 5 7 11 13 17 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} 0_{+} 1_{k} 1_{−} 0 551  1^{2} 1_{k} 19 19 1^{2} 1_{−} 29∗ 29∗  2 2 2
C_{2} 26  2 2 2 2 2 2
2. 1_{k} 4_{+} 1_{k} 1_{−} 0 -21 -55 1^{3} 4_{+} 1 1^{2} 1_{k} 19 19 1^{2} 1_{−} 29∗ 29∗ 3 2 2 2 3 3
S_{3} 6  3 3 2 2 2 2 2 2

Discriminant class: −87

∞ 3 19 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 19  Over p = 29  2 5 7 11 13 17 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 2 0_{+} ∗ ∗ 1^{2} 1_{−} 29∗ 29∗  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 2 0_{+} ∗ ∗ 1^{2} 1_{−} 29∗ 29∗ 3 2 3 3 3 3 2 2
S_{3} 2 2 3 2 2 2 3

Discriminant class: −19

∞ 3 19 29 a_{1} a_{2}  a_{3} Over p = 3  Over p = 19  Over p = 29 2 5 7 11 13 17 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
5. 1_{k} 0_{+} 1_{i} 0_{+} -1  5  2 0_{+} ∗ ∗ 1^{2} 1_{i} 19∗ 19∗ 2 0_{+} ∗ ∗ 2  2  2
C_{2}  2 2  2 2  2
6. 1_{k} 4_{+} 1_{i} 0_{+} 0  6  -5 1^{3} 4_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 2 0_{+} ∗ ∗ 2  3 3 2 3 3 2
S_{3}  2 2 3 3 2 2 3 2
7. 1_{k} 0_{+} 1_{i} 2_{+} -1 10 44 2 0_{+} ∗ ∗ 1^{2} 1_{i} 19∗ 19∗ 1^{3} 2_{+} ∗ 2 3 3 2 2
S_{3}  2 2 3 3 2 2 3 2
8. 1_{k} 4_{+} 1_{i} 2_{+} 0 -261 -2088 1^{3} 4_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 1^{3} 2_{+} ∗  2 3 3 3 2 3 3 2
S_{3} 3  2 2  2 2 3 2
9. 1_{k} 4_{+} 1_{i} 2_{+} 0 -174 -1769 1^{3} 4_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 1^{3} 2_{+} ∗  2 3  3 2 3 3 2
S_{3} 30  2 2 3 3 2 2  2

Discriminant class: −3

∞ 3 19 29 a_{1} a_{2} a_{3} Over p = 3 Over p = 19 Over p = 29 2 5 7 11 13 17 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
10. 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
11. 1_{k} 5_{i} 0_{+} 0_{+} 0 0  -3 1^{3} 5_{i} 3∗  3 0_{+} 1  2 0_{+} ∗ ∗ 2 2 3 2 3 2 2 3
S_{3}  3 2 3 2 2 2
12. 1_{k} 1_{i} 2_{+} 0_{+} -1 -6 -12 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1 2 0_{+} ∗ ∗ 2 2 3 2 3 2 2 3
S_{3} 3  3 2 3 2 2 2 3 3
13. 1_{k} 3_{i} 0_{+} 2_{+} 0 0 -29 1^{3} 3_{i} 3∗ 3 0_{+} 1 1^{3} 2_{+} ∗ 2 2 2 3 2 2
S_{3}  2 3 2 2 2 3 3

Discriminant class: −3 (continued)

∞ 3 19 29 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 19 Over p = 29 2 5 7 11 13 17 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
14. 1_{k} 5_{i} 2_{+} 0_{+} 0 0 -171 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 0_{+} ∗ ∗ 2 2 3 2 3 2 2
S_{3} 6  2  2 2 2 3 3
15. 1_{k} 5_{i} 2_{+} 0_{+} 0 0  -57 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 0_{+} ∗ ∗ 2 2  2  2 2 3
S_{3} 6  3 2 3 2 2 2 3 3
16. 1_{k} 5_{i} 0_{+} 2_{+} 0 0  -87 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 2 3 2 3 2 2 3
S_{3}  3 2  2 2 2 3 3
17. 1_{k} 5_{i} 0_{+} 2_{+} 0 0 -261 1^{3} 5_{i} 3∗  3 0_{+} 1  1^{3} 2_{+} ∗  2 2 3 2  2 2 3
S_{3}  3 2 3 2 2 2 3 3
18. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -10469 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 2 2 3
S_{3} 24  3 2 3 2 2 2  3
19. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -551 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2  2 2 3
S_{3} 3  3 2  2 2 2 3
20. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -4959 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 2 2
S_{3} 3  2 3 2 2 2 3
21. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -31407 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2  2 2
S_{3} 9  2 3 2 2 2  3
22. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -47937 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 3 2 2 3
S_{3} 3  3 2  2 2 2  3
23. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1653 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 3 2 2 3
S_{3} 18  3 2 3 2 2 2 3

Discriminant class: 1

∞ 3 19 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 19 Over p = 29 2 5 7 11 13 17 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
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 -6 7 3 0_{+} 1  1^{3} 2_{+} 1  3 0_{+} 1  3 3  3 3 3
C_{3}  3 3 3 3 3 3 3
26. 0_{+} 4_{+} 2_{+} 0_{+} 0 -57 19 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3  3 3 3 3 3 3
C_{3} 3  3  3  3
27. 0_{+} 4_{+} 2_{+} 0_{+} 0 -57 152 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 0_{+} 1  3 3 3  3  3
C_{3} 3  3  3 3 3 3

Discriminant class: 29

∞ 3 19 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 19  Over p = 29 2 5 7 11 13 17 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
28. 0_{+} 0_{+} 0_{+} 1_{+} 0 -29  2 0_{+} ∗ ∗ 2 0_{+} ∗ ∗ 1^{2} 1_{+} 29 29 2  2  2  2
C_{2}  2 2 2 2  2
29. 0_{+} 4_{+} 0_{+} 1_{+} 0 -12 -13 1^{3} 4_{+} ∗  2 0_{+} ∗ ∗ 1^{2} 1_{+} 29 29 2 3 3 2  2 3 2
S_{3}  2 2 2 2 3 3 2 3

Discriminant class: 57

∞ 3 19 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 19  Over p = 29 2 5 7 11 13 17 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
30. 0_{+} 1_{k} 1_{i} 0_{+} 0 -57  1^{2} 1_{k} 3 3 1^{2} 1_{i} 19∗ 19∗  2  2 2 2 2 2
C_{2}  2  2  2

Discriminant class: 1653

∞ 3 19 29 a_{1}  a_{2} a_{3}  Over p = 3  Over p = 19  Over p = 29 2 5 7 11 13 17 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
31. 0_{+} 1_{i} 1_{k} 1_{+} 0 -1653  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{k} 19 19 1^{2} 1_{+} 29 29 2 2  2  2
C_{2} 2  2 2  2 2
32. 0_{+} 5_{i} 1_{k} 1_{+} 0 -90 -321 1^{3} 5_{i} 3∗  1^{2} 1_{k} 19 19 1^{2} 1_{+} 29 29 2 2 3 3 2  2 3
S_{3} 2  3 2 2  3 3 2 2