Quintic Fields Ramifying Above {2, 5}

Discriminant class: −10

∞ 2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 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} 9_{i} 5_{+} 0 0 0 1^{4} 9_{i} −2∗ −∗ 1^{5} 5_{+} 5∗  4 5 3 5 32 5 3 32
S_{5}  -15 -4  4 3 5 2 5 3 5 4
2. 1_{k} 5_{i} 7_{+} 0 -5 0 1^{3} 2_{+} ∗  1^{5} 7_{+} 5∗  4 3 5 5 4 5 22 4
S_{5}  0 -10  1^{2} 3_{i} −2 −2  32 5 3 4 5 22 3 4

Discriminant class: −5

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 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
3. 1_{k} 10_{k} 5_{−}  0  0 0 1^{4} 10_{k} −∗ −2 1^{5} 5_{−} 5  5 5 32 4 2 4 5 3
S_{5}  -10 -8  4 32 3 3 22 32 32 3
4. 1_{k} 4_{k} 9_{−}  0  0 0 2^{2} 4_{k} −∗  ∗ 1^{5} 9_{−} 5  3 5 4 32 4 2 3 5
S_{5}  -125 -200  4 2 5 5 5 4 4 5
5. 1_{k} 10_{k} 7_{−}  0 -10 0 1^{4} 10_{k} −∗ −2∗ 1^{5} 7_{−} 5  3 3 32 4 32 4 3 5
S_{5}  -125 80  32 4 22 22 3 32 4 5
6. 1_{k} 6_{k} 9_{−}  0 -25 -50 1^{2} 3_{−} 2∗  2∗ 1^{5} 9_{−} 5  5 5 32 4 32 2 3 22
S_{5}  0 160  1^{2} 3_{i} −2 −2  4 32 22 3 5 4 4 3
7. 1_{k} 6_{k} 9_{−}  0  0 0 2^{2} 6_{k} −∗  ∗ 1^{5} 9_{−} 5  5 5 4 32 32 4 22 5
S_{5}  -25 -20  32 4 5 5 3 32 4 3
8. 1_{k} 6_{k} 9_{−}  0  0 0 1^{2} 3_{+} 2  2 1^{5} 9_{−} 5  5 3 32 32 4 32 5 3
S_{5}  -100 160  1^{2} 3_{k} −2∗ −2∗  4 2 22 22 5 32 32 22
9. 1_{k} 10_{k} 9_{−}  0  0 0 1^{4} 10_{k} −∗ −2∗ 1^{5} 9_{−} 5  3 3 4 4 32 4 5 5
S_{5}  -50 -40  4 4 5 5 22 2 2 3
10. 1_{k} 10_{k} 9_{−}  0 -50 0 1^{4} 10_{k} −∗ −2∗ 1^{5} 9_{−} 5  22 3 32 2 4 4 5 5
S_{5}  2  275 -520  32 32 3 5 5 32 4 3
11. 1_{k} 10_{k} 9_{−}  0  0 0 1^{4} 10_{k} −∗ −2 1^{5} 9_{−} 5  5 22 4 32 32 4 5 5
S_{5}  -250 200  4 4 22 5 5 4 32 5

Discriminant class: −2

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 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
12. 1_{k} 11_{i} 6_{+}  0 -10 -20 1^{4} 11_{i} −2 −2∗ 1^{5} 6_{+} ∗  5 32 5 32 5 5 32 4
S_{5}  35 -4  2 4 5 5 4 2 5 4
13. 1_{k} 11_{i} 6_{+}  0 -10 -20 1^{4} 11_{i} −2  2 1^{5} 6_{+} ∗  5 32 22 32 5 3 4 4
S_{5}  15  4  4 2 3 22 4 4 5 32
14. 1_{k} 11_{i} 8_{+}  0 -50 100 1^{4} 11_{i} −2  2 1^{5} 8_{+} ∗  22 4 5 32 5 3 32 32
S_{5}  375 -900  32 32 5 3 4 4 3 32
15. 1_{k} 11_{i} 8_{+}  0 10 -140 1^{4} 11_{i} −2 −2∗ 1^{5} 8_{+} ∗  3 4 22 4 5 3 32 4
S_{5}  -215 -52  2 4 3 3 32 32 3 4

Discriminant class: −1

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 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
16. 1_{k} 4_{i} 8_{+} 0 -10 30 1^{3} 2_{+} ∗  1^{5} 8_{+} 1  4 32 4 5 3 4 4 5
S_{5}  -40 16  1^{2} 2_{i} −∗ −∗  4 22 5 4 32 3 4 3
17. 1_{k} 10_{i} 6_{+} 0 -10 0 1^{4} 10_{i} −1 2 1^{5} 6_{+} 1  32 4 4 3 5 4 32 5
S_{5} 15 -8 4 22 5 32 4 3 4 22
18. 1_{k} 10_{i} 6_{+} 0 -10 0 1^{4} 10_{i} −1 2∗ 1^{5} 6_{+} 1  4 4 4 3 3 32 32 3
S_{5}  -25 -24  32 3 5 4 2 5 32 3

Discriminant class: 1

∞ 2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 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
19. 2_{−} 4_{−} 6_{+} 0 -5 0 1^{2} 2_{i} −∗ −∗ 1^{5} 6_{+} 1  5 5 22 22 22 22 5 5
D_{5}  10 -4  1^{2} 2_{i} −∗ −∗  22 22 5 5 5 22 22 5
20. 2_{−} 6_{−} 6_{+} 0  0 0 1^{2} 3_{k} −2∗ −2∗ 1^{5} 6_{+} 1  22 5 5 5 22 5 5 22
D_{5}  -5 -12  1^{2} 3_{k} −2∗ −2∗  22 5 5 22 5 5 5 22
21. 2_{−} 4_{−} 8_{+} 0 10 -10 1^{2} 2_{i} −1 −1 1^{5} 8_{+} 1  22 22 22 5 5 22 22 5
D_{5} 5  -15 -18  1^{2} 2_{i} −1 −1  22 5 5 22 22 5 22 5
22. 2_{−} 6_{−} 8_{+} 0 10 -40 1^{2} 3_{i} −2 −2 1^{5} 8_{+} 1  5 22  22 5 5 22 22
D_{5}  -15 88  1^{2} 3_{i} −2 −2  22 22 5 5 22 22 5 22
23. 2_{−} 6_{−} 6_{+} 0  0 0 1^{4} 6_{−} 1  1^{5} 6_{+} 1  5 3 3 5 3 5 22 3
A_{5} 3  20 -16  5 22 3 3 5 5 5 5
24. 2_{−} 6_{−} 8_{+} 0 10 -40 2^{2} 6_{−} 1  {∗ \atop 2} 1^{5} 8_{+} 1  5 3 3 3 5 5 3 3
A_{5} 15  60 -32  3 5 3 3 3 22 22 5
25. 2_{−} 6_{−} 8_{+} 0 -10 20 1^{4} 6_{−} 1  1^{5} 8_{+} 1  5 5 5 3 3 5 5 22
A_{5} 3  110 -116  22 3 22 22 5 22 3 5
26. 2_{−} 6_{−} 8_{+} 0 10 -10 1^{4} 6_{−} 1  1^{5} 8_{+} 1  3 5 22 5 22 5 3 5
A_{5} 15  35 -18  3 3 22 5 5 3 5 3
27. 2_{−} 8_{−} 8_{+} 0 -10 -20 1^{4} 8_{−} 1 {−2∗ \atop −1} 1^{5} 8_{+} 1  3 5 3 3 5 22 5 3
A_{5} 3  10 216  3 5 22 5 3 3 3 3

Discriminant class: 2

∞  2 5 a_{1}  a_{2} a_{3}  Over p = 2  Over p = 5 3 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
28. 0_{+} 11_{+} 6_{+} 0 -20  0 1^{4} 11_{+} 2 2 1^{5} 6_{+} ∗  4 5 4 4 22 4 5 4
F_{5}  60 -32  5 4 22 4 22 4 4 4
29. 2_{−} 11_{−} 6_{+} 0  0  0 1^{4} 11_{−} 2 2 1^{5} 6_{+} ∗  4 22 4 4 22 4 22 4
F_{5}  120  64  22 4 22 4 5 4 4 4
30. 2_{−} 11_{−} 6_{+} 0 -10 -20 1^{4} 11_{−} 2 2 1^{5} 6_{+} ∗  4 22 4 4 22 4 22 4
F_{5}  -25 -20  22 4 22 4 5 4 4 4
31. 2_{−} 11_{−} 8_{+} 0 -10 -180 1^{4} 11_{−} 2 2 1^{5} 8_{+} ∗  4 5 4 4  4 5 4
F_{5}  935 -1156  22 4 22 4 22 4 4 4
32. 2_{−} 11_{−} 8_{+} 0  10 -60 1^{4} 11_{−} 2 2 1^{5} 8_{+} ∗  4 5 4 4  4 5 4
F_{5}  -15 -148  22 4 22 4 22 4 4 4
33. 2_{−} 5_{−} 6_{+} 0  5  0 1^{3} 2_{+} ∗  1^{5} 6_{+} ∗  32 3 4 4 5 32 5 4
S_{5}  10  -2  1^{2} 3_{−} 2∗ 2∗  5 4 3 32 3 32 4 4

Discriminant class: 2 (continued)

∞  2 5 a_{1}  a_{2} a_{3}  Over p = 2  Over p = 5 3 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
34. 2_{−} 5_{−} 8_{+} 0 -15 -10 1^{3} 2_{+} ∗  1^{5} 8_{+} ∗  4 22 32 4 5 32 3 4
S_{5}  60  92  1^{2} 3_{−} 2∗ 2∗  3 4 5 32 5 32 4 2
35. 2_{−} 11_{−} 6_{+} 0  0  0 1^{4} 11_{−} 2 −2 1^{5} 6_{+} ∗  4 3 4 32 5 32 3 32
S_{5}  -30 -64  3 4 22 32 3 32 2 4
36. 2_{−} 9_{−} 8_{+} 0  10 -40 1^{4} 9_{−} 2 −1 1^{5} 8_{+} ∗  4 5 2 32 3 4 3 2
S_{5}  10 -192  3 4 3 2 22 4 4 4
37. 2_{−} 11_{−} 8_{+} 0  10 -60 1^{4} 11_{−} 2 2 1^{5} 8_{+} ∗  32 5 4 32 3 4 5 4
S_{5}  135 -108  5 32 3 4 5 32 4 4

Discriminant class: 5

∞ 2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 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_{−} 4_{+} 5_{−} 0  0  0 1^{5} 4_{+} ∗  1^{5} 5_{−} 5  4 4 5 4 4 22 4 22
F_{5}  0 -2  5 4 5 4 4 4 22 5
39. 2_{−} 6_{+} 5_{−} 0  5  0 2^{2} 6_{+} ∗ ∗ 1^{5} 5_{−} 5  4 4 22 4 4 5 4 5
F_{5}  5 -2  5 4 5 4 4 4 5 22
40. 2_{−} 6_{+} 7_{−} 0  5  0 2^{2} 6_{+} ∗ ∗ 1^{5} 7_{−} 5  4 4 22 4 4  4 5
F_{5}  5 -6  5 4 5 4 4 4 5 22
41. 2_{−} 4_{+} 9_{−} 0  0  0 1^{5} 4_{+} ∗  1^{5} 9_{−} 5  4 4  4 4 22 4 22
F_{5}  0 -10  5 4 5 4 4 4 22 5
42. 2_{−} 4_{+} 9_{−} 0  0  0 1^{5} 4_{+} ∗  1^{5} 9_{−} 5  4 4 5 4 4 22 4 22
F_{5} 2  0 -40  5 4  4 4 4 22
43. 2_{−} 4_{+} 9_{−} 0  0  0 1^{5} 4_{+} ∗  1^{5} 9_{−} 5  4 4 5 4 4 22 4 22
F_{5}  0 -80  5 4 5 4 4 4 22 5
44. 0_{+} 4_{−} 9_{−} 0 -25  0 2^{2} 4_{−} ∗ ∗ 1^{5} 9_{−} 5  4 4 22 4 4 5 4 22
F_{5}  125 -50  22 4 5 4 4 4 5 5
45. 2_{−} 4_{+} 9_{−} 0  0  0 1^{5} 4_{+} ∗  1^{5} 9_{−} 5  4 4 5 4 4 22 4 22
F_{5}  0 -20  5 4 5 4 4 4 22 5
46. 2_{−} 8_{+} 5_{−} 0  0 -20 1^{4} 8_{+} ∗  1^{5} 5_{−} 5  32 32 5 32 4 3 4 5
S_{5}  -10 -8  22 2 3 4 4 4 5 5
47. 2_{−} 4_{+} 7_{−} 0  0 -10 1^{4} 4_{+} ∗  1^{5} 7_{−} 5  32 4 3 4 2 5 32 3
S_{5}  -10 -16  22 32 5 4 32 4 3 22
48. 2_{−} 2_{+} 9_{−} 0  0  0 1^{3} 2_{+} ∗  1^{5} 9_{−} 5  32 4 22 32 32 5 4 5
S_{5}  25 -10  3 4 3 2 32 4 3 5
49. 2_{−} 8_{+} 7_{−} 0  0 -20 1^{4} 8_{+} ∗  1^{5} 7_{−} 5  4 2 3 32 4 3 4 3
S_{5}  -10 -48  5 32 5 4 32 2 3 5
50. 2_{−} 8_{+} 7_{−} 0 -10 -20 1^{4} 8_{+} ∗ −∗ 1^{5} 7_{−} 5  4 32 5 2 32 3 4 3
S_{5}  110 -104  3 32 5 32 2 32 3 3
51. 2_{−} 4_{+} 9_{−} 0  0  0 1^{4} 4_{+} ∗  1^{5} 9_{−} 5  2 32 3 32 4 5 4 3
S_{5}  -25 -30  5 32 3 32 32 4 5 5
52. 2_{−} 6_{+} 9_{−} 0  0  0 1^{4} 6_{+} ∗ −1 1^{5} 9_{−} 5  32 4 3 4 4 3 4 3
S_{5}  100 160  3 4 22 32 4 32 22 5
53. 2_{−} 8_{+} 9_{−} 0  0  0 1^{4} 8_{+} ∗  1^{5} 9_{−} 5  2 4 3 32 32 3 4 5
S_{5} 5  200 -480  3 32 5 4 2 32 22 22
54. 2_{−} 8_{+} 9_{−} 0  0 -100 1^{4} 8_{+} ∗  1^{5} 9_{−} 5  32 2 5 32 32 5 2 22
S_{5}  50 560  5 4 3 32 4 2 5 3

Discriminant class: 5 (continued)

∞ 2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 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
55. 2_{−} 8_{+} 9_{−} 0  0 -100 1^{4} 8_{+} ∗  1^{5} 9_{−} 5  4 32 5 2 2 22 32 3
S_{5}  450 -640  5 4 5 4 2 4 5 3

Discriminant class: 10

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  3 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
56. 2_{−} 11_{−} 3_{+} -2  6 -8 1^{4} 11_{−} 2∗ 2∗ 1^{4} 3_{+} 5∗ 5∗ 5 4 4 5 4 4 4 4
F_{5}  9 -14  5 5 22 22 4 22 4 4
57. 2_{−} 11_{−} 5_{+} 0 -10 20 1^{4} 11_{−} 2∗ 2∗ 1^{5} 5_{+} 5∗  5 4 4  4 4 4 4
F_{5}  5  15 20  5 5 22 22 4 22 4 4
58. 2_{−} 11_{−} 5_{+} 0 10 20 1^{4} 11_{−} 2∗ 2∗ 1^{5} 5_{+} 5∗  22 4 4 22 4 4 4 4
F_{5} -15 -36 5 22 22 5 4 5 4 4
59. 2_{−} 11_{−} 7_{+} 0 10 -60 1^{4} 11_{−} 2∗ 2∗ 1^{5} 7_{+} 5∗  22 4 4 22 4 4 4 4
F_{5}  95 -44  22 22 5 4 5 4 4
60. 2_{−} 3_{−} 7_{+} 0 -5 -5 3 0_{+} 1  1^{5} 7_{+} 5∗  5 32 4 22 4 32 4 2
S_{5}  0 -4  1^{2} 3_{−} 2∗ 2∗  3 3 22 3 4 5 4 4
61. 2_{−} 9_{−} 5_{+} 0  0 0 1^{4} 9_{−} 2∗ −1 1^{5} 5_{+} 5∗  5 32 4 3 32 2 4 32
S_{5}  5 -4  5 5 5 3 2 3 4 4
62. 2_{−} 11_{−} 5_{+} 0  0 0 1^{4} 11_{−} 2∗ −2∗ 1^{5} 5_{+} 5∗  3 32 32 3 2 4 4 4
S_{5}  10 -16  5 3 5 3 4 5 32 32
63. 2_{−} 11_{−} 7_{+} 0 -10 -20 1^{4} 11_{−} 2∗ −2∗ 1^{5} 7_{+} 5∗  3 32 4 5 4 4 32 32
S_{5} 145 -132 3 22 22 22 32 5 4 4
64. 2_{−} 11_{−} 7_{+} 0 -10 -20 1^{4} 11_{−} 2∗ 2∗ 1^{5} 7_{+} 5∗  5 2 32 3 32 32 32 4
S_{5}  95 -52  22 5 22 3 4 5 32 2