Quartic Fields Unramified Outside of {∞, 2, 17}

Discriminant class: −34

∞  2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17 3 5 7 11 13 19 23 29
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 1_{k} 11_{i} 1_{+}  0 8 0 1^{4} 11_{i} −2 2 1^{2} 1_{+} 17 17 4 22  4 4 22 22 22
D_{4}  -34  22 2 22 2 4 22 22
2. 1_{k} 11_{i} 1_{+}  0 -8 0 1^{4} 11_{i} −2 2 1^{2} 1_{+} 17 17 4 22 22 4 4 22  22
D_{4}  -34  22 22 2 22 2 4 22 22
3. 1_{k} 3_{i} 3_{+} -1 -6 18 1^{2} 3_{i} −2 −2 1^{4} 3_{+} 17 17 4 22 22 4 2  22 22
D_{4}  2  -16  22 22 4 22 2 2 22 22
4. 1_{k} 5_{i} 3_{+}  0 -17 0 1^{2} 3_{+} 2  2 1^{4} 3_{+} 17 17 4 22 22 4 2 22 22 22
D_{4}  2  -34  1^{2} 2_{i} −1 −1  22 22 4  2 2  22
5. 1_{k} 11_{i} 3_{+}  0 0 0 1^{4} 11_{i} −2 2 1^{4} 3_{+} 17 17 2  22 2 4 22 22
D_{4} 16  -136  22 22 4 22 2 4 22 22
6. 1_{k} 11_{i} 3_{+}  0 0 0 1^{4} 11_{i} −2 2 1^{4} 3_{+} 17 17 2 22 22 2 4 22 22 22
D_{4} 8 -34 22 4 22 2 4 22

Discriminant class: −17

∞  2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17 3 5 7 11 13 19 23 29
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
7. 1_{k} 10_{i} 1_{+} 0 -2 0 1^{4} 10_{i} −1 2 1^{2} 1_{+} 17 17 22 4 22 22 22 4  4
D_{4} -17 4 2 4 2 22 4 4
8. 1_{k} 10_{i} 1_{+} 0 2 0 1^{4} 10_{i} −1 2 1^{2} 1_{+} 17 17 22 4  22 22 4 22 4
D_{4} -17 22 4 2 4 2 22 4 4
9. 1_{k} 2_{i} 3_{+} 0 0 0 1^{2} 2_{i} −1 −1 1^{4} 3_{+} 17 17 22 4 22 22 22 2 22 4
D_{4} 2 -17 22 4 4 2 2 22 2 4
10. 1_{k} 6_{i} 3_{+} 0 0 0 1^{2} 3_{i} −2 −2 1^{4} 3_{+} 17 17 22 4 22 22  2 22 4
D_{4} 2 -68 1^{2} 3_{+} 2 2 22 4 4 2 2 2 4
11. 1_{k} 10_{i} 3_{+} 0 -34 0 1^{4} 10_{i} −1 2 1^{4} 3_{+} 17 17 22 2 22  22 4 22 2
D_{4} 8 -17 22 2 4 4 2 22 4 2
12. 1_{k} 10_{i} 3_{+} 0 34 0 1^{4} 10_{i} −1 2 1^{4} 3_{+} 17 17  2 22 22 22 4 22 2
D_{4} 8 -17 22 2 4 4 2 22 4 2

Discriminant class: −2

∞  2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  3 5 7 11 13 19 23 29
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
13. 1_{k} 11_{i} 0_{+}  0 0 0 1^{4} 11_{i} −2  2 2 0_{+} ∗  ∗ 22 4 2 22 4 22 2 4
D_{4}  -2  2 0_{+} ∗  ∗ 2 4 22 22 2 4 22 4
14. 1_{k} 3_{k} 2_{−}  0 -3 0 1^{2} 3_{k} −2∗ −2∗ 2^{2} 2_{−} 1  V 22 4 4 22 2 22 4 4
D_{4}  -2  2 0_{+} ∗  ∗  4 4 22  2 2  4
15. 1_{k} 5_{k} 2_{−}  0 3 0 1^{2} 3_{−} 2∗  2∗ 2^{2} 2_{−} 1  V 22 4 4 22 2  4 4
D_{4}  -2  1^{2} 2_{i} −∗ −∗  4 4 22 22 2 2 22 4
16. 1_{k} 11_{i} 2_{+}  0 -8 0 1^{4} 11_{i} −2  2 1^{2} 1_{+} 17 17 22 2 4  4 22 4 2
D_{4}  2  -18  1^{2} 1_{+} 17 17 4 2 22 22 2 4 22 2

Discriminant class: −2 (continued)

∞  2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  3 5 7 11 13 19 23 29
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
17. 1_{k} 11_{i} 2_{+}  0 8 0 1^{4} 11_{i} −2  2 1^{2} 1_{+} 17 17  2 4 22 4 22 4 2
D_{4}  2  -18  1^{2} 1_{+} 17 17 4 2 22 22 2 4 22 2
18. 1_{k} 11_{i} 2_{+}  0 0 0 1^{4} 11_{i} −2  2 1^{2} 1_{−} 17∗ 17∗ 22 4 2 22 4 22 2 4
D_{4}  2  -578  1^{2} 1_{−} 17∗ 17∗ 2 4  22 2 4 22 4

Discriminant class: −1

∞  2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  3 5 7 11 13 19 23 29
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
19. 1_{k} 10_{i} 0_{+}  0 -2 0 1^{4} 10_{i} −1  2 2 0_{+} ∗  ∗ 4 22 2 4 22 4 2 22
D_{4}  -1  2 0_{+} ∗  ∗ 2 22  4 2 22 4 22
20. 1_{k} 2_{i} 2_{+}  1 -2 1 1^{2} 2_{i} −∗ −∗ 1^{2} 1_{+} 17 17 4 22 4 4 22 2 4 22
D_{4}  1  2 0_{+} ∗  ∗ 1^{2} 1_{+} 17 17 4 22 22 2 2  2 22
21. 1_{k} 6_{i} 2_{+}  0 2 0 1^{2} 3_{−} 2∗  2∗ 1^{2} 1_{+} 17 17 4 22 4 4  2 4 22
D_{4}  -16  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{+} 17 17 4 22 22 2 2 22 2 22
22. 1_{k} 10_{i} 2_{+}  0 34 0 1^{4} 10_{i} −1  2 1^{2} 1_{−} 17∗ 17∗ 4 22 2 4 22 4 2 22
D_{4}  2  -289  1^{2} 1_{−} 17∗ 17∗ 2 22 22 4 2 22 4 22
23. 1_{k} 10_{k} 2_{−}  0 -6 0 1^{4} 10_{k} −1  2 2^{2} 2_{−} 1  V 2  4 2 22 4 4 22
D_{4}  2  -25  4 22 22 4 2 22 4 22
24. 1_{k} 10_{k} 2_{−}  0 -10 0 1^{4} 10_{k} −1  2 2^{2} 2_{−} 1  V 2 22 4 2 22 4 4
D_{4}  2  -9  4  22 4 2 22 4

Discriminant class: 1

∞ 2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17 3 5 7 11 13 19 23 29
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
25. 2_{−} 8_{−} 0_{+} 0 0 0 1^{4} 8_{−} 1 {−2 \atop −1}  22 22 22 22 22 22 22 22
V_{4} 1 22 22 22 22 22 22 22
26. 2_{−} 4_{−} 2_{+} 0 9 0 1^{2} 2_{i} −1 −1 1^{2} 1_{+} 17 17 22 22 22 22  22 22 22
V_{4} 2 16 1^{2} 2_{i} −1 −1 1^{2} 1_{+} 17 17 22 22 22 22 22 22 22
27. 0_{+} 6_{+} 2_{+} -2 -11 12 1^{2} 3_{+} 2  2 1^{2} 1_{+} 17 17 22 22 22 22 22 22 22 22
V_{4} 2 1^{2} 3_{+} 2 2 1^{2} 1_{+} 17 17 22 22 22 22 22 22 22
28. 2_{−} 6_{−} 2_{+} 2 -3 -4 1^{2} 3_{i} −2 −2 1^{2} 1_{+} 17 17 22 22 22 22 22  22 22
V_{4} 2  38  1^{2} 3_{i} −2 −2 1^{2} 1_{+} 17 17 22 22 22  22 22  22
29. 2_{−} 8_{−} 2_{+} 0 16 0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{2} 1_{+} 17 17 22 22  22 22 22  22
V_{4} 8  81  1^{2} 1_{+} 17 17  22 22 22 22 22 22 22
30. 2_{−} 8_{−} 2_{+} 0 0 0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{2} 1_{+} 17 17 22  22 22 22 22 22
V_{4} 8  289  1^{2} 1_{+} 17 17 22  22 22 22 22 22
31. 2_{−} 8_{−} 2_{+} 0 -16 0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{2} 1_{+} 17 17  22 22  22 22 22 22
V_{4} 8  81  1^{2} 1_{+} 17 17 22 22 22 22 22 22 22 22

Discriminant class: 2

∞  2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  3 5 7 11 13 19 23 29
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
32. 2_{−} 11_{−} 0_{+} 0 4 0 1^{4} 11_{−} 2  2  4 4  4 4 4  4
C_{4}  2  22 4 22 4 22 4 4 4
33. 0_{+} 11_{+} 0_{+} 0 -4 0 1^{4} 11_{+} 2  2  4 4 22 4 4 4 22 4
C_{4}  2  4 22 4  4 4 4
34. 0_{+} 11_{+} 2_{+} 0 -68 0 1^{4} 11_{+} 2  2 1^{2} 1_{+} 17 17 4 4  4 4 4  4
C_{4}  2  578  1^{2} 1_{+} 17 17 22 4  4  4 4 4
35. 2_{−} 11_{−} 2_{+} 0 68 0 1^{4} 11_{−} 2  2 1^{2} 1_{+} 17 17 4 4 22 4 4 4 22 4
C_{4} 8 578 1^{2} 1_{+} 17 17 4 4 22 4 4 4
36. 2_{−} 9_{−} 0_{+} 0 -2 0 1^{4} 9_{−} 2 −1 2 0_{+} ∗  ∗ 4 2 22 4 2 4 22 2
D_{4} 2 2 0_{+} ∗ ∗ 22 2 4 22 2 4 2
37. 2_{−} 11_{−} 0_{+} 0 0 0 1^{4} 11_{−} 2 −2 2 0_{+} ∗  ∗ 2 4 22 2 4 2 22 4
D_{4}  2  2 0_{+} ∗  ∗ 22 4 22 2 22 4 2 4
38. 2_{−} 3_{+} 2_{−} 1 0 2 1^{2} 3_{+} 2  2 2^{2} 2_{−} 1  V 4 4 22 4 2 2 22 4
D_{4}  4  22 4 22 2  2 2 4
39. 0_{+} 5_{−} 2_{−} 0 -5 0 1^{2} 3_{i} −2 −2 2^{2} 2_{−} 1  V 4 4 22 4 2 2 22 4
D_{4} 2 1^{2} 2_{i} −1 −1 22 4 22 2 22 2 2 4
40. 2_{−} 9_{−} 2_{+} 0 -2 0 1^{4} 9_{−} 2 −1 1^{2} 1_{+} 17 17 2 4 22 2 2 4  4
D_{4} 4 18 1^{2} 1_{+} 17 17 4 22 4 22 2 4 4
41. 2_{−} 9_{−} 2_{+} 0 -34 0 1^{4} 9_{−} 2 −1 1^{2} 1_{−} 17∗ 17∗ 4 2 22 4 2 4 22 2
D_{4} 4 578 1^{2} 1_{−} 17∗ 17∗ 22 2 22 4 22 2 4 2
42. 2_{−} 9_{−} 2_{+} 0 2 0 1^{4} 9_{−} 2 −1 1^{2} 1_{+} 17 17 2 4  2 2 4 22 4
D_{4}  4  18  1^{2} 1_{+} 17 17 22 4 22 4 22 2 4 4
43. 0_{+} 11_{−} 2_{−} 0 -12 0 1^{4} 11_{−} 2  2 2^{2} 2_{−} 1  V 2 2 22 2 4 4 22 2
D_{4}  2  2  22 2 22 4 22 4 4 2
44. 2_{−} 11_{+} 2_{−} 0 12 0 1^{4} 11_{+} 2  2 2^{2} 2_{−} 1  V 2 2 22 2 4 4 22 2
D_{4}  8  2  22 2 22 4  4 4 2
45. 2_{−} 11_{−} 2_{+} 0 8 0 1^{4} 11_{−} 2 −2 1^{2} 1_{+} 17 17 4 2  4 4 2 22 2
D_{4} 4 50 1^{2} 1_{+} 17 17 2 22 2 22 4 2 2
46. 2_{−} 11_{−} 2_{+} 0 -8 0 1^{4} 11_{−} 2 −2 1^{2} 1_{+} 17 17 4 2 22 4 4 2  2
D_{4} 4 50 1^{2} 1_{+} 17 17 22 2 22 2 22 4 2 2
47. 2_{−} 11_{−} 2_{+} 0 0 0 1^{4} 11_{−} 2 −2 1^{2} 1_{−} 17∗ 17∗ 2 4 22 2 4 2 22 4
D_{4}  4  578  1^{2} 1_{−} 17∗ 17∗ 22 4  2 22 4 2 4

Discriminant class: 17

∞ 2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  3 5 7 11 13 19 23 29
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
48. 0_{+} 0_{+} 3_{+} 1 -6 -1 2 0_{+} ∗  ∗ 1^{4} 3_{+} 17 17 4 4 4 4  22 4 4
C_{4}  1  2 0_{+} ∗  ∗  4 4 4 22  22 22 4
49. 2_{−} 4_{−} 3_{+} 0 17 0 1^{2} 2_{i} −∗ −∗ 1^{4} 3_{+} 17 17 4 4 4 4  4 4
C_{4} 4  68  1^{2} 2_{i} −∗ −∗  4 4 4  22 22  4
50. 0_{+} 6_{+} 3_{+} 0 -34 0 1^{2} 3_{−} 2∗  2∗ 1^{4} 3_{+} 17 17 4 4 4 4 22  4 4
C_{4} 2  272  1^{2} 3_{−} 2∗  2∗  4 4 4  4
51. 2_{−} 6_{−} 3_{+} 0 34 0 1^{2} 3_{k} −2∗ −2∗ 1^{4} 3_{+} 17 17 4 4 4 4 22 22 4 4
C_{4} 4 272 1^{2} 3_{k} −2∗ −2∗ 4 4 4 22 22 22 4
52. 2_{−} 4_{−} 1_{+} 0 1 2 2^{2} 4_{−} 1  {−1 \atop −∗} 1^{2} 1_{+} 17 17 4 2 4 4 22 22 4 2
D_{4}  1  4 2 2 22 22  22 2

Discriminant class: 17 (continued)

∞ 2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  3 5 7 11 13 19 23 29
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
53. 2_{−} 6_{+} 1_{−} -2 5 -4 1^{2} 3_{+} 2  2 1^{2} 1_{−} 17∗ 17∗ 4 4 2 4 22 22 2 4
D_{4} 2 1^{2} 3_{+} 2 2 2 0_{+} ∗ ∗ 2 4 2 22 22 22 4
54. 2_{−} 6_{+} 1_{−} 2 1 2 2^{2} 6_{+} 1 {−2 \atop −2∗} 1^{2} 1_{−} 17∗ 17∗ 2 4 4 2 22 22 4 4
D_{4} 3 2 0_{+} ∗ ∗ 4 4 2 22 22 4
55. 0_{+} 8_{−} 1_{−} 0 -6 -4 1^{4} 8_{−} 1  {−2 \atop −1} 1^{2} 1_{−} 17∗ 17∗ 4 4 2 4 22 22 2 4
D_{4}  2  2 0_{+} ∗  ∗ 2 4 2 22 22 22 22 4
56. 2_{−} 8_{+} 1_{−} 0 -2 4 1^{4} 8_{+} 1  {−2 \atop −∗} 1^{2} 1_{−} 17∗ 17∗ 2 4 4 2 22  4 4
D_{4} 2  6  2 0_{+} ∗  ∗ 4 4 2 22 22 22 22 4
57. 2_{−} 8_{−} 1_{+} 0 8 0 1^{4} 8_{−} 1 {−2∗ \atop −1} 1^{2} 1_{+} 17 17 4 2 4 4  22 4 2
D_{4} 2 17 4 2 2 22 22 22 22 2
58. 2_{−} 4_{−} 3_{+} 2 10 -8 1^{2} 2_{i} −1 −1 1^{4} 3_{+} 17 17 2 4 2 2 22 22 2 4
D_{4} 8 16 1^{2} 2_{i} −1 −1 2 4 4 22 22 22 22 4
59. 2_{−} 6_{−} 3_{+} 2 -7 -8 1^{2} 3_{i} −2 −2 1^{4} 3_{+} 17 17 4 2 2 4 22  2 2
D_{4} 8 50 1^{2} 3_{i} −2 −2 2 2 4 22 22 22 22 2
60. 2_{−} 6_{−} 3_{+} 2 -7 -42 2^{2} 6_{−} 1  {∗ \atop 2} 1^{4} 3_{+} 17 17 2 2 4 2 22 22 4 2
D_{4} 4 135 4 2 4 22 22 22 2
61. 0_{+} 8_{+} 3_{+} 0 -34 68 1^{4} 8_{+} 1 {−2∗ \atop −∗} 1^{4} 3_{+} 17 17 2 2 4 2 22 22 4 2
D_{4} 4 -34 4 2 4 22 22 22 22 2
62. 2_{−} 8_{−} 3_{+} 0 34 -68 1^{4} 8_{−} 1  {−2 \atop −1} 1^{4} 3_{+} 17 17 4 2 2 4 22 22 2 2
D_{4} 16  34  2 2 4  22 22  2
63. 2_{−} 8_{−} 3_{+} 0 0 0 1^{4} 8_{−} 1  {−2 \atop −1} 1^{4} 3_{+} 17 17 2 4 2 2  22 2 4
D_{4} 16  17  2 4 4 22 22  22 4

Discriminant class: 34

∞  2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  3 5 7 11 13 19 23 29
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
64. 0_{+} 11_{+} 3_{+} 0 -68 0 1^{4} 11_{+} 2  2 1^{4} 3_{+} 17 17 22  4 22 4 4 4 22
C_{4}  4  306  4  4 4  4 4
65. 2_{−} 11_{−} 3_{+} 0 68 0 1^{4} 11_{−} 2  2 1^{4} 3_{+} 17 17  4  4 4 4 22
C_{4} 104  306  4  4 4 22 4 4
66. 0_{+} 11_{+} 3_{+} 0 -68 0 1^{4} 11_{+} 2  2 1^{4} 3_{+} 17 17  22 4  4 4 4
C_{4}  4  850  4 22 4 4  4 4 22
67. 2_{−} 11_{−} 3_{+} 0 68 0 1^{4} 11_{−} 2  2 1^{4} 3_{+} 17 17 22 22 4 22 4 4 4
C_{4} 8 850 4 22 4 4 22 4 4 22
68. 2_{−} 9_{+} 1_{−} 0 -6 0 1^{4} 9_{+} 2 −1 1^{2} 1_{−} 17∗ 17∗ 22 22 4 22 2 4 4
D_{4} 2 34 2 0_{+} ∗ ∗ 4 2 4 22 2 4
69. 2_{−} 9_{+} 1_{−} 0 -10 0 1^{4} 9_{+} 2 −1 1^{2} 1_{−} 17∗ 17∗ 22  4 22 2 4 4 22
D_{4} 2 34 2 0_{+} ∗ ∗ 4 22 2 4 22 2 4 22
70. 2_{−} 11_{−} 1_{+} 0 -8 0 1^{4} 11_{−} 2 −2 1^{2} 1_{+} 17 17  22 4 22 4 2 4 22
D_{4} 2 34 4 22 2 2 22 4 2 22
71. 0_{+} 11_{−} 1_{−} 0 -12 0 1^{4} 11_{−} 2  2 1^{2} 1_{−} 17∗ 17∗ 22 22 2 22 4 4 2 22
D_{4} 34 2 0_{+} ∗ ∗ 2 22 2 4 22 4 4 22
72. 2_{−} 11_{+} 1_{−} 0 12 0 1^{4} 11_{+} 2  2 1^{2} 1_{−} 17∗ 17∗ 22 22 2 22 4 4 2 22
D_{4} 4 34 2 0_{+} ∗ ∗ 2 22 2 4 4 4 22
73. 2_{−} 11_{−} 1_{+} 0 8 0 1^{4} 11_{−} 2 −2 1^{2} 1_{+} 17 17 22 22 4  4 2 4 22
D_{4}  2  34  4 22 2 2 22 4 2 22

Discriminant class: 34 (continued)

∞  2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  3 5 7 11 13 19 23 29
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
74. 2_{−} 3_{−} 3_{+} -1 11 -16 1^{2} 3_{−} 2∗  2∗ 1^{4} 3_{+} 17 17 22 22 4 22 2 2 4 22
D_{4} 2 52 2 0_{+} ∗ ∗ 4 22 4 2 2 2 22
75. 0_{+} 5_{+} 3_{+} 0 -17 0 1^{2} 3_{k} −2∗ −2∗ 1^{4} 3_{+} 17 17 22 22 4 22 2 2 4 22
D_{4} 2 34 1^{2} 2_{i} −∗ −∗ 4 22 4 2 22 2 2 22
76. 2_{−} 9_{−} 3_{+} 0 -34 0 1^{4} 9_{−} 2 −1 1^{4} 3_{+} 17 17  22 2 22 2 4 2 22
D_{4} 16 306 2 22 4 4 22 2 4 22
77. 2_{−} 9_{−} 3_{+} 0 34 0 1^{4} 9_{−} 2 −1 1^{4} 3_{+} 17 17 22 22 2  2 4 2 22
D_{4} 16 306 2 22 4 4 22 2 4 22
78. 2_{−} 11_{−} 3_{+} 0 0 0 1^{4} 11_{−} 2 −2 1^{4} 3_{+} 17 17 22  2 22 4 2 2
D_{4} 32 34 2 22 4 2 22 4 2 22
79. 2_{−} 11_{−} 3_{+} 0 0 0 1^{4} 11_{−} 2 −2 1^{4} 3_{+} 17 17 22 22 2 22 4 2 2 22
D_{4} 16  136  2  4 2 22 4 2