Quintic Fields Unramified Outside of {∞, 5, 17}

Discriminant class: 1

∞ 5 17 a_{1} a_{2} a_{3} Over p = 5  Over p = 17 2 3 7 11 13 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. 0_{+} 8_{+} 0_{+} 0 -10 -5 1^{5} 8_{+} 1  5 0_{+} 1  5 5  5 5 5 5 5
C_{5}  10 -1  5 5 5  5 5 5 5

Discriminant class: 5

∞ 5 17 a_{1}  a_{2} a_{3} Over p = 5  Over p = 17 2 3 7 11 13 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
2. 2_{−} 7_{−} 0_{+}  0  5  0 1^{5} 7_{−} 5  4 0_{+} ∗ ∗ 4 4 4 5 4 22 4 22
F_{5}  5  -1  5 4 5 4 4 4 22 5
3. 2_{−} 5_{−} 2_{+}  0  5  0 1^{5} 5_{−} 5  2^{2} 2_{+} ∗ ∗ 4 4 4 22 4 22 4 5
F_{5}  5  -8  22 4 22 4 4 4 22 22
4. 2_{−} 9_{−} 0_{+}  0  0  0 1^{5} 9_{−} 5  4 0_{+} ∗ ∗ 4 4 4 5 4 22 4 22
F_{5}  0  -5  4 5 4 4 4 22 5
5. 2_{−} 7_{−} 2_{+}  0  5  0 1^{5} 7_{−} 5  2^{2} 2_{+} ∗ ∗ 4 4 4 22 4 22 4 5
F_{5}  5  -9  22 4 22 4 4 4 22 22
6. 2_{−} 5_{−} 4_{+}  0  0  0 1^{5} 5_{−} 5  1^{5} 4_{+} ∗  4 4 4 5 4 22 4 22
F_{5}  0  -17  5 4 5 4 4 4 22 5
7. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  4 4 4 5 4 22 4 22
F_{5}  0 -425  5 4 5 4 4 4 22 5
8. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  4 4 4 5 4 22 4 22
F_{5} 2  0  -85  5 4  4 4 4 22 5
9. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  4 4 4 5 4 22 4 22
F_{5}  0 -1445  5 4 5 4 4 4 22 5
10. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  4 4 4  4 22 4 22
F_{5} 2  0 -10625  5 4 5 4 4 4 22
11. 2_{−} 5_{−} 4_{+}  0  0  0 1^{5} 5_{−} 5  1^{5} 4_{+} ∗  32 2 32 22 32 5 4 5
S_{5}  -85 -153  3 4 3 4 2 4 3 22
12. 2_{−} 9_{−} 2_{+}  0  0 -50 1^{5} 9_{−} 5  1^{3} 2_{+} ∗  2 32 4 5 32 3 4 3
S_{5}  -25 -230  5 32 3 32 4 2 5 5
13. 2_{−} 9_{−} 2_{+}  0  0 -50 1^{5} 9_{−} 5  2^{2} 2_{+} ∗ ∗ 4 4 2 3 2 3 32 5
S_{5}  100  -65  5 2 5 4 4 4 5 5
14. 2_{−} 9_{−} 2_{+}  0  25 -125 1^{5} 9_{−} 5  1^{3} 2_{+} ∗  32 4 2 5 32 5 4 5
S_{5}  250 -420  5 4 3 4 2 4 3 5
15. 2_{−} 7_{−} 4_{+}  0  0 -85 1^{5} 7_{−} 5  1^{5} 4_{+} ∗  32 4 32 3 4 5 2 22
S_{5}  -85  -51  5 32 5 4 32 4 3 22
16. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  32 32 4 5 32 22 4 5
S_{5}  2125 -8075  22 4 22 4 4 2 3 3

Discriminant class: 17

∞ 5 17 a_{1}  a_{2} a_{3} Over p = 5  Over p = 17  2 3 7 11 13 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. 0_{+} 8_{+} 3_{+}  0 -65 -110 1^{5} 8_{+} ∗  1^{4} 3_{+} 17 17 22 4 4 4 5 22 4 4
F_{5}  535 792  4 4 4 22 5 22 22 4
18. 2_{−} 2_{+} 3_{−}  0  1 -2 1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  5 4 32 4 5 3 32 4
S_{5}  -2  -3  1^{2} 1_{−} 17∗ 17∗ 4 4 32 5 3 5 22 4
19. 2_{−} 6_{+} 1_{−}  0  0 -5 1^{5} 6_{+} ∗  1^{2} 1_{−} 17∗ 17∗ 5 32 4 4 22 5 4 4
S_{5}  0  5  2 0_{+} ∗  ∗ 32 32 32 3 3 3 3 2
20. 2_{−} 6_{+} 3_{−}  0 -15 -75 1^{5} 6_{+} ∗  1^{3} 2_{+} ∗  22 32 32 4 5 5 4 32
S_{5}  -110 -89  1^{2} 1_{−} 17∗ 17∗ 4 2 4 3 5 3 3 2
21. 2_{−} 8_{+} 3_{−}  0  15 45 1^{5} 8_{+} ∗  1^{3} 2_{+} ∗  22 4 32 4 22 5 2 4
S_{5}  60 239  1^{2} 1_{−} 17∗ 17∗ 4 32 2 5 3 3 22 4
22. 2_{−} 8_{+} 3_{−}  0  35 -15 1^{5} 8_{+} ∗  1^{3} 2_{+} ∗  3 4 4 2 5 3 4 2
S_{5} 2  185 -1102  1^{2} 1_{−} 17∗ 17∗ 32 4 4 22 22 5 5 32

Discriminant class: 85

∞ 5 17 a_{1} a_{2} a_{3}  Over p = 5  Over p = 17  2 3 7 11 13 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
23. 2_{−} 3_{+} 3_{−} -2 2 3 1^{4} 3_{+} 5∗ 5∗ 1^{4} 3_{−} 17∗ 17∗ 4 5 5 4 4 5 22 4
F_{5} -3 4 4 22 4 4 4 4 4
24. 2_{−} 5_{+} 3_{−} 0 10 -5 1^{5} 5_{+} 5∗  1^{4} 3_{−} 17∗ 17∗ 4 22 22 4 4 5 5 4
F_{5} 2  -15 18  4 5 4 4 4 4 5 4
25. 2_{−} 7_{+} 3_{−} 0 -5 -30 1^{5} 7_{+} 5∗  1^{4} 3_{−} 17∗ 17∗ 4 22 22 4 4  5 4
F_{5}  -25 541  4 5 4 4 4 4 5 4
26. 2_{−} 7_{+} 3_{−} 0 20 -25 1^{5} 7_{+} 5∗  1^{4} 3_{−} 17∗ 17∗ 4 5 5 4 4  22 4
F_{5} 5  25 10  4 22 4 4 4 4 5 4
27. 2_{−} 7_{+} 3_{−} 0 -10 -20 1^{5} 7_{+} 5∗  1^{4} 3_{−} 17∗ 17∗ 32 5 5 32 2 5 5 4
S_{5} 3  -15 421  4 5 32 32 4 32 5 4