Quintic Fields Unramified Outside of {∞, 3, 5}

Discriminant class: −15

∞ 3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 23 29
G h  a_{4} a_{5}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 1_{k} 5_{i} 5_{+} 0 -5 0 1^{3} 5_{i} 3∗  1^{5} 5_{+} 5∗  5 32 4 2 22 5 3 32
S_{5}  0 -9  2 0_{+} ∗ ∗  22 32 4 4 3 3 32 5
2. 1_{k} 5_{i} 5_{+} 0 0 -5 1^{3} 4_{+} ∗  1^{5} 5_{+} 5∗  5 4 4 4 3 3 5 4
S_{5}  0 3  1^{2} 1_{i} 3∗ 3∗  5 4 32 32 5 22 32 5
3. 1_{k} 3_{i} 7_{+} 0 -5 5 1^{3} 3_{i} 3∗  1^{5} 7_{+} 5∗  3 4 32 32 5 5 22 4
S_{5}  0 -16  2 0_{+} ∗ ∗  5 32 32 4 5 3 32 5
4. 1_{k} 5_{i} 7_{+} 0 -15 -15 1^{3} 4_{+} ∗  1^{5} 7_{+} 5∗  3 32 32 32 3 22 5 4
S_{5}  45 144  1^{2} 1_{i} 3∗ 3∗  5 4 4 2 22 5 4 3

Discriminant class: −3

∞ 3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 23 29
G h  a_{4} a_{5}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
5. 1_{k} 5_{i} 6_{+} 0 -10 -10 1^{3} 5_{i} 3∗  1^{5} 6_{+} ∗  4 5 4 5 32 22 4 32
S_{5}  30 43  22 22 32 3 32 4 32 3
6. 1_{k} 5_{i} 6_{+} 0 0 -10 1^{3} 4_{+} 1  1^{5} 6_{+} ∗  4 3 32 5 4 3 4 32
S_{5}  0 15  1^{2} 1_{i} 3∗ 3∗  5 5 4 5 32 2 32 3
7. 1_{k} 3_{i} 8_{+} 0 -10 5 1^{4} 3_{i} 3∗ 3 1^{5} 8_{+} ∗  32 5 32 5 32 22 4 4
S_{5}  -15 31  3 3 4 3 4 4 32 5
8. 1_{k} 5_{i} 8_{+} 0 -10 -5 1^{3} 5_{i} 3∗  1^{5} 8_{+} ∗  32 22 32 5 4 22 32 4
S_{5}  -15 89  5 22 2 3 4 4 32 3

Discriminant class: 1

∞ 3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 23 29
G h  a_{4} a_{5}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
9. 0_{+} 0_{+} 8_{+} 0 -10 -5 5 0_{+} 1  1^{5} 8_{+} 1  5  5 5 5 5 5 5
C_{5}  10 -1  5 5 5  5 5 5 5
10. 2_{−} 2_{−} 6_{+} 0 0 -5 1^{2} 1_{k} 3 3 1^{5} 6_{+} 1  5 22 22 22 5 5 5 22
D_{5} 0 -3 1^{2} 1_{k} 3 3 5 22 22 22 5 5 22 5
11. 2_{−} 2_{−} 8_{+} 0 10 -15 1^{2} 1_{i} 3∗ 3∗ 1^{5} 8_{+} 1  22 5 22 5 22 5 22 22
D_{5}  10 -12  1^{2} 1_{i} 3∗ 3∗  5 5 22 5 22 22 22 5
12. 2_{−} 6_{−} 6_{+} 0 15 -5 1^{3} 5_{i} 3∗  1^{5} 6_{+} 1  3 5 5 22 3 3 22 5
A_{5}  60 -96  1^{2} 1_{i} 3∗ 3∗  5 3 22 5 5 5 3 5
13. 2_{−} 6_{−} 6_{+} 0 15 -25 1^{3} 5_{i} 3∗  1^{5} 6_{+} 1  5 5 5 5 5 3 5 22
A_{5}  15 -3  1^{2} 1_{i} 3∗ 3∗  3 5 5 22 22 3 3 5
14. 2_{−} 4_{−} 8_{+} 0 0 25 1^{3} 3_{k} 3  1^{5} 8_{+} 1  5 3 5 5 3 3 22 5
A_{5}  0 -75  1^{2} 1_{k} 3 3  3 22 5 5 3 5 5 3
15. 2_{−} 4_{−} 8_{+} 0 0 -25 1^{3} 3_{i} 3∗  1^{5} 8_{+} 1  5 5 3 22 3 3 3 5
A_{5} 5  0 -75  1^{2} 1_{i} 3∗ 3∗  22 5 3 22 5 5 22 3
16. 2_{−} 6_{−} 8_{+} 0 15 -20 1^{3} 5_{i} 3∗  1^{5} 8_{+} 1  3 22 5 3 5 5 5 22
A_{5} 5  60 -24  1^{2} 1_{i} 3∗ 3∗  3 5 3 3 3 22 22 22

Discriminant class: 1 (continued)

∞ 3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 23 29
G h  a_{4} a_{5}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
17. 2_{−} 6_{−} 8_{+} 0 -15 -15 1^{3} 5_{i} 3∗  1^{5} 8_{+} 1  5 3 22 3 5 22 5 5
A_{5}  135 63  1^{2} 1_{i} 3∗ 3∗  3 3 3 5 5  5 3

Discriminant class: 5

∞ 3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 23 29
G h  a_{4} a_{5}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
18. 2_{−} 0_{+} 7_{−}  0  5  0 4 0_{+} ∗ ∗ 1^{5} 7_{−} 5  4 4 5 4 4 22 4 22
F_{5}  5 -1  5 4 5 4 4 4 22 5
19. 2_{−} 4_{+} 5_{−}  0  0  0 1^{5} 4_{+} ∗  1^{5} 5_{−} 5  4 4 5 4 4 22 4 22
F_{5}  0 -3  5 4  4 4 4 22 5
20. 2_{−} 0_{+} 9_{−}  0  0  0 4 0_{+} ∗ ∗ 1^{5} 9_{−} 5  4 4 5 4 4 22 4 22
F_{5}  0 -5  4 5 4 4 4 22 5
21. 2_{−} 4_{+} 9_{−}  0  0  0 1^{5} 4_{+} ∗  1^{5} 9_{−} 5  4 4  4 4 22 4 22
F_{5}  0 -45  5 4 5 4 4 4 22 5
22. 2_{−} 4_{+} 9_{−}  0  0  0 1^{5} 4_{+} ∗  1^{5} 9_{−} 5  4 4 5 4 4 22 4 22
F_{5}  0 -405  5 4 5 4 4 4 22 5
23. 2_{−} 4_{+} 9_{−}  0  0  0 1^{5} 4_{+} ∗  1^{5} 9_{−} 5  4 4 5 4 4 22 4 22
F_{5}  0 -75  5 4 5 4 4 4 22
24. 2_{−} 4_{+} 9_{−}  0  0  0 1^{5} 4_{+} ∗  1^{5} 9_{−} 5  4 4 5 4 4 22 4 22
F_{5}  0 -15  5 4 5 4 4 4 22 5
25. 2_{−} 6_{+} 5_{−}  0  0 -5 1^{3} 5_{i} 3∗  1^{5} 5_{−} 5  32 4 3 4 4 5 4 3
S_{5}  15 33  1^{2} 1_{k} 3 3  3 4 5 4 32 32 5 22
26. 2_{−} 6_{+} 7_{−}  0  0 -20 1^{3} 5_{i} 3∗  1^{5} 7_{−} 5  32 2 5 4 4 5 4 3
S_{5}  15 -3  1^{2} 1_{k} 3 3  22 32 3 4 4 4 5 22
27. 2_{−} 6_{+} 7_{−}  0  0 -5 1^{3} 5_{i} 3∗  1^{5} 7_{−} 5  4 4 3 32 4 22 4 5
S_{5}  15 -12  1^{2} 1_{k} 3 3  22 4 5 4 2 32 3 3
28. 2_{−} 6_{+} 7_{−}  0 -15 -50 1^{3} 5_{i} 3∗  1^{5} 7_{−} 5  4 2 5 32 32 5 2 3
S_{5}  -75 -45  1^{2} 1_{k} 3 3  3 32 5 4 4 4 3 22
29. 2_{−} 4_{+} 9_{−}  0  0 75 1^{3} 4_{+} 1  1^{5} 9_{−} 5  32 4 5 32 32 3 32 5
S_{5} 2  225 135  2 0_{+} ∗ ∗  22 32 22 32 4 32 5 5
30. 2_{−} 4_{+} 9_{−}  0  0 -75 1^{3} 3_{i} 3∗  1^{5} 9_{−} 5  4 4 5 2 4 5 4 3
S_{5}  225 -180  1^{2} 1_{k} 3 3  22 2 22 4 32 32 3 5
31. 2_{−} 4_{+} 9_{−}  0 -25  0 1^{3} 4_{+} ∗  1^{5} 9_{−} 5  2 4 5 4 32 5 4 5
S_{5}  0 -600  3 4 5 4 4 4 3 22
32. 2_{−} 4_{+} 9_{−}  0 -25  0 1^{3} 4_{+} ∗  1^{5} 9_{−} 5  4 2 22 4 32 22 4 22
S_{5}  225 -285  3 2 5 32 4 32 5 5
33. 2_{−} 4_{+} 9_{−}  0  0  0 1^{5} 4_{+} ∗  1^{5} 9_{−} 5  32 2 3 32 4 5 2 3
S_{5}  75 -105  22 2 22 4 4 2 5 5
34. 2_{−} 6_{+} 9_{−}  0  0 -125 1^{3} 5_{i} 3∗  1^{5} 9_{−} 5  4 2 5 4 4 3 4 5
S_{5}  375 -300  1^{2} 1_{k} 3 3  5 4 5 4 32 32 22 22
35. 2_{−} 6_{+} 9_{−}  0  0 -50 1^{3} 5_{i} 3∗  1^{5} 9_{−} 5  32 4 22 4 2 3 4 5
S_{5}  -75 -165  1^{2} 1_{k} 3 3  3 32 22 32 4 4 5 22
36. 2_{−} 6_{+} 9_{−}  0  0 -100 1^{3} 5_{i} 3∗  1^{5} 9_{−} 5  32 4 3 2 32 5 4 5
S_{5}  -75 -15  1^{2} 1_{k} 3 3  5 2  32 2 32 5 3
37. 2_{−} 6_{+} 9_{−}  0  0 -50 1^{3} 5_{i} 3∗  1^{5} 9_{−} 5  32 4 5 4 2 22 4 22
S_{5}  375 285  1^{2} 1_{k} 3 3  22 32 3 32 2 32 5 5

Discriminant class: 5 (continued)

∞ 3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 23 29
G h  a_{4} a_{5}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
38. 2_{−} 6_{+} 9_{−}  0  0 -25 1^{3} 5_{i} 3∗  1^{5} 9_{−} 5  2 32 3 32 32 5 4 22
S_{5}  -300 -960  1^{2} 1_{k} 3 3  22 2 3 32 32 32 5 3