Quartic Fields Ramifying Above {2, 41}

Discriminant class: −82

∞  2 41 a_{1} a_{2} a_{3}  Over p = 2  Over p = 41 3 5 7 11 13 17 19 23
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
1. 1_{k} 11_{i} 1_{+}  0 -8 0 1^{4} 11_{i} −2 2 1^{2} 1_{+} 41 41 4 4  4 22 2 4 2
D_{4}  2  -82  22 2 4 22  22 22 4
2. 1_{k} 11_{i} 1_{+}  0 8 0 1^{4} 11_{i} −2 2 1^{2} 1_{+} 41 41 4 4 22 4 22 2 4 2
D_{4} 2 -82 22 2 4 22 22 22 22 4
3. 1_{k} 3_{i} 3_{+}  1 -15 59 1^{2} 3_{i} −2 −2 1^{4} 3_{+} 41 41 4 2 22 4 22 4 4 2
D_{4}  2  -86  22 2 2  22 22 22 2
4. 1_{k} 5_{i} 3_{+}  0 -41 0 1^{2} 3_{+} 2  2 1^{4} 3_{+} 41 41 4 2 22 4 22 4 4 2
D_{4}  4  -82  1^{2} 2_{i} −1 −1  22 2 2 22 22 22  2
5. 1_{k} 11_{i} 3_{+}  0 0 0 1^{4} 11_{i} −2 2 1^{4} 3_{+} 41 41 2 4 22 2  4 2 2
D_{4} 32  -328  22 2 4 22 22  22 4
6. 1_{k} 11_{i} 3_{+}  0 0 0 1^{4} 11_{i} −2 2 1^{4} 3_{+} 41 41 2 4 22 2 22 4 2 2
D_{4} 64 -82 2 4 22 22 22 22 4

Discriminant class: −41

∞  2 41 a_{1} a_{2} a_{3}  Over p = 2  Over p = 41  3 5 7 11 13 17 19 23
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 29 31 37 43 47 53 59 61
7. 1_{k} 10_{k} 1_{−}  0 -6 0 1^{4} 10_{k} −1  2 1^{2} 1_{−} 41∗ 41∗ 22 22  22 4 2 22 2
D_{4}  2  -41  2 0_{+} ∗  ∗ 4 2 22 4 22 4 4 22
8. 1_{k} 10_{k} 1_{−}  0 6 0 1^{4} 10_{k} −1  2 1^{2} 1_{−} 41∗ 41∗ 22 22 22 22 4 2 22 2
D_{4}  2  -41  2 0_{+} ∗  ∗ 4 2 22 4  4 4 22
9. 1_{k} 2_{i} 3_{+}  0 0 0 1^{2} 2_{i} −∗ −∗ 1^{4} 3_{+} 41 41 22  22 22 4 4 22 2
D_{4}  2  -41  2 0_{+} ∗  ∗  4 2 22 2 22 4 2 22
10. 1_{k} 6_{i} 3_{+}  0 0 0 1^{2} 3_{k} −2∗ −2∗ 1^{4} 3_{+} 41 41 22 22 22 22 4 4 22 2
D_{4}  4  -164  1^{2} 3_{−} 2∗  2∗  4 2  2 22 4 2
11. 1_{k} 10_{i} 3_{+}  0 82 0 1^{4} 10_{i} −1  2 1^{4} 3_{+} 41 41 22 22 22  2 4 22 2
D_{4} 16  -369  2 2 22 4 22 2 4 22
12. 1_{k} 10_{i} 3_{+}  0 -82 0 1^{4} 10_{i} −1  2 1^{4} 3_{+} 41 41  22 22 22 2 4  2
D_{4} 16  -369  2 2 22 4 22 2 4 22

Discriminant class: −2

∞  2 41  a_{1} a_{2} a_{3}  Over p = 2  Over p = 41  3 5 7 11 13 17 19 23
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 29 31 37 43 47 53 59 61
13. 1_{k} 3_{k} 2_{−}  0 3 0 1^{2} 3_{k} −2∗ −2∗ 2^{2} 2_{−} 1  V 22 2 4 22 4 22 22 2
D_{4}  -8  2 0_{+} ∗  ∗  4 2 2 22 4 4 22 2
14. 1_{k} 5_{k} 2_{−}  0 -3 0 1^{2} 3_{−} 2∗  2∗ 2^{2} 2_{−} 1  V 22 2 4 22 4 22 22 2
D_{4}  2  -8  1^{2} 2_{i} −∗ −∗  4 2 2  4 4  2
15. 1_{k} 11_{i} 2_{+}  0 -16 0 1^{4} 11_{i} −2  2 1^{2} 1_{+} 41 41  4 4  2 22 22 2
D_{4}  8  -18  1^{2} 1_{+} 41 41 2 2 4 22 4 2 22 4
16. 1_{k} 11_{i} 2_{+}  0 0 0 1^{4} 11_{i} −2  2 1^{2} 1_{−} 41∗ 41∗ 22 4 2 22 4  22 2
D_{4}  4  -3362  1^{2} 1_{−} 41∗ 41∗ 4 2 4 22 2 4 22 4

Discriminant class: −2 (continued)

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

Discriminant class: −1

∞  2 41  a_{1} a_{2} a_{3}  Over p = 2  Over p = 41 3 5 7 11 13 17 19 23
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
18. 1_{k} 2_{i} 2_{+}  0 5 0 1^{2} 2_{i} −1 −1 1^{2} 1_{+} 41 41 4 22 4 4 22 22 4 2
D_{4}  -4  1^{2} 1_{+} 41 41 22 2 22 2 4 22 2
19. 1_{k} 6_{i} 2_{+}  0 -10 0 1^{2} 3_{i} −2 −2 1^{2} 1_{+} 41 41 4  4 4 22 22 4 2
D_{4}  2  -16  1^{2} 3_{+} 2  2 1^{2} 1_{+} 41 41 22 2  2 4 22 2 22
20. 1_{k} 10_{i} 2_{+}  0 -2 0 1^{4} 10_{i} −1 2 1^{2} 1_{+} 41 41 2 22 4 2  22 2 2
D_{4}  8  -81  1^{2} 1_{+} 41 41 22 2 22 4 4 22 4 22
21. 1_{k} 10_{i} 2_{+}  0 18 0 1^{4} 10_{i} −1 2 1^{2} 1_{+} 41 41 2 22 4 2 22 22 2 2
D_{4}  8  -1  1^{2} 1_{+} 41 41  2 22 4 4  4 22
22. 1_{k} 10_{i} 2_{+}  0 82 0 1^{4} 10_{i} −1 2 1^{2} 1_{+} 41 41 4 22 2 4 22  4 2
D_{4}  8  -1681  1^{2} 1_{+} 41 41 22 2 22 4 2 22 4 22

Discriminant class: 1

∞ 2 41 a_{1} a_{2} a_{3}  Over p = 2  Over p = 41 3 5 7 11 13 17 19 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
23. 2_{−} 4_{−} 2_{+}  0 21 0 1^{2} 2_{i} −1 −1 1^{2} 1_{+} 41 41 22  22 22 22 22 22 22
V_{4} 4  100  1^{2} 2_{i} −1 −1 1^{2} 1_{+} 41 41 22 22  22 22 22 22
24. 2_{−} 6_{−} 2_{+}  2 -15 -16 1^{2} 3_{i} −2 −2 1^{2} 1_{+} 41 41 22 22 22 22 22 22 22 22
V_{4} 2  146  1^{2} 3_{i} −2 −2 1^{2} 1_{+} 41 41 22 22 22  22 22  22
25. 0_{+} 6_{+} 2_{+}  2 -23 -24 1^{2} 3_{+} 2  2 1^{2} 1_{+} 41 41 22 22 22 22 22 22 22
V_{4} 2  62  1^{2} 3_{+} 2  2 1^{2} 1_{+} 41 41 22  22 22 22 22 22 22
26. 2_{−} 8_{−} 2_{+}  0 40 0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{2} 1_{+} 41 41 22 22  22 22 22 22 22
V_{4} 16  441  1^{2} 1_{+} 41 41 22 22 22 22  22 22 22
27. 2_{−} 8_{−} 2_{+}  0 -40 0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{2} 1_{+} 41 41  22 22  22 22  22
V_{4} 16  441  1^{2} 1_{+} 41 41 22 22 22 22 22 22 22 22
28. 2_{−} 8_{−} 2_{+}  0 0 0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{2} 1_{+} 41 41 22 22 22 22  22 22 22
V_{4} 8  1681  1^{2} 1_{+} 41 41  22 22 22 22  22 22

Discriminant class: 2

∞  2 41 a_{1} a_{2} a_{3}  Over p = 2  Over p = 41  3 5 7 11 13 17 19 23
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 29 31 37 43 47 53 59 61
29. 2_{−} 11_{−} 2_{+}  0 164 0 1^{4} 11_{−} 2  2 1^{2} 1_{−} 41∗ 41∗ 4 4 22 4 4 22 4
C_{4} 20  3362  1^{2} 1_{−} 41∗ 41∗ 4 22 4 4  4 4 4
30. 0_{+} 11_{+} 2_{+}  0 -164 0 1^{4} 11_{+} 2  2 1^{2} 1_{−} 41∗ 41∗ 4 4  4 4 22 4 22
C_{4}  4  3362  1^{2} 1_{−} 41∗ 41∗ 4  4 4 22 4 4 4

Discriminant class: 2 (continued)

∞  2 41 a_{1} a_{2} a_{3}  Over p = 2  Over p = 41  3 5 7 11 13 17 19 23
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 29 31 37 43 47 53 59 61
31. 0_{+} 3_{−} 2_{−}  0 -7 0 1^{2} 3_{−} 2∗  2∗ 2^{2} 2_{−} 1  V 4 2 22 4 4 22 4 22
D_{4}  2  2 0_{+} ∗  ∗  4 22 2 2 22 4 2 2
32. 2_{−} 5_{+} 2_{−}  0  7 0 1^{2} 3_{k} −2∗ −2∗ 2^{2} 2_{−} 1  V 4 2 22 4 4 22 4
D_{4}  2  2  1^{2} 2_{i} −∗ −∗  4  2 2 22 4 2 2
33. 2_{−} 9_{+} 2_{−}  0  6 0 1^{4} 9_{+} 2 −1 2^{2} 2_{−} 1  V 2 2 22 2 4 22 2 22
D_{4} 16  50  4 22 2 4  4 4 2
34. 2_{−} 9_{+} 2_{−}  0 -6 0 1^{4} 9_{+} 2 −1 2^{2} 2_{−} 1  V 2 2  2 4 22 2 22
D_{4} 16  50  4 22 2 4 22 4 4 2
35. 2_{−} 9_{−} 2_{+}  0 -82 0 1^{4} 9_{−} 2 −1 1^{2} 1_{+} 41 41 4 2 22 4 2  4 22
D_{4}  8  3362  1^{2} 1_{+} 41 41 2 22 2 4 22 2 4 2
36. 2_{−} 11_{−} 2_{+}  0  8 0 1^{4} 11_{−} 2 −2 1^{2} 1_{+} 41 41 4 4 22 4 2 22 4 22
D_{4}  8  98  1^{2} 1_{+} 41 41 2 22 4 2 22 2 2 4
37. 2_{−} 11_{−} 2_{+}  0 -8 0 1^{4} 11_{−} 2 −2 1^{2} 1_{+} 41 41 4 4  4 2 22 4 22
D_{4}  8  98  1^{2} 1_{+} 41 41 2 22 4 2  2 2 4
38. 0_{+} 11_{+} 2_{+}  0 -20 0 1^{4} 11_{+} 2  2 1^{2} 1_{+} 41 41 2 4 22 2 2 22 2
D_{4}  8  18  1^{2} 1_{+} 41 41 2 22 4 4 22 2 4 4
39. 2_{−} 11_{−} 2_{+}  0  0 0 1^{4} 11_{−} 2 −2 1^{2} 1_{−} 41∗ 41∗ 2 4 22 2 4  2 22
D_{4}  4  3362  1^{2} 1_{−} 41∗ 41∗ 4 22 4 2 22 4 2 4
40. 2_{−} 11_{−} 2_{+}  0 20 0 1^{4} 11_{−} 2  2 1^{2} 1_{+} 41 41 2 4 22 2 2 22 2 22
D_{4} 24  18  1^{2} 1_{+} 41 41 2  4 4 22 2 4 4

Discriminant class: 41

∞ 2 41 a_{1} a_{2} a_{3}  Over p = 2  Over p = 41  3 5 7 11 13 17 19 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 29 31 37 43 47 53 59 61
41. 2_{−} 4_{−} 3_{+}  0 41  0 1^{2} 2_{i} −∗ −∗ 1^{4} 3_{+} 41 41 4 22 4 4 4 4 4 22
C_{4} 4  164  1^{2} 2_{i} −∗ −∗  4 22  4 4 22 22
42. 2_{−} 6_{−} 3_{+}  0 82  0 1^{2} 3_{k} −2∗ −2∗ 1^{4} 3_{+} 41 41 4  4 4 4 4 4 22
C_{4} 20  656  1^{2} 3_{k} −2∗ −2∗  4 22 22 22 4 4
43. 0_{+} 6_{+} 3_{+}  0 -82  0 1^{2} 3_{−} 2∗  2∗ 1^{4} 3_{+} 41 41 4  4 4 4 4 4
C_{4} 4  656  1^{2} 3_{−} 2∗  2∗  4  22  4 4 22
44. 2_{−} 4_{−} 1_{+}  0 3  2 1^{2} 2_{i} −1 −1 1^{2} 1_{+} 41 41 4 22 4 4 2 2 4 22
D_{4}  2  1^{2} 2_{i} −1 −1  2 22 22 22 4 2 22
45. 0_{+} 6_{−} 1_{−}  -2 -3  6 2^{2} 6_{−} 1  {∗ \atop 2} 1^{2} 1_{−} 41∗ 41∗ 4 22 2 4 4 2 4 22
D_{4}  -1  2 0_{+} ∗  ∗ 4 22 22 22 2 4 22 22
46. 2_{−} 6_{+} 1_{−}  2 3  2 2^{2} 6_{+} 1 {−2 \atop −2∗} 1^{2} 1_{−} 41∗ 41∗ 2 22 4 2 4 2 2 22
D_{4}  3  2 0_{+} ∗  ∗ 4 22 22 22 4 4 22 22
47. 2_{−} 8_{+} 1_{−}  0 6 -4 1^{4} 8_{+} 1 {−2∗ \atop −∗} 1^{2} 1_{−} 41∗ 41∗ 4 22 2 4 4 2 4
D_{4} 2  14  2 0_{+} ∗  ∗ 4  22 22 2 4 22 22
48. 2_{−} 8_{−} 1_{+}  0 8  0 1^{4} 8_{−} 1  {−2 \atop −1} 1^{2} 1_{+} 41 41 4  4 4 2 2 4 22
D_{4} 2  41  2 22  22 4 2 22 22
49. 2_{−} 8_{+} 1_{−}  0 4 -8 1^{4} 8_{+} 1  {−2 \atop −∗} 1^{2} 1_{−} 41∗ 41∗ 2 22 4 2 4 2 2 22
D_{4} 2  9  2 0_{+} ∗  ∗ 4 22 22  4 4  22
50. 2_{−} 4_{−} 3_{+}  0 0  0 2^{2} 4_{−} 1  {−1 \atop −∗} 1^{4} 3_{+} 41 41 2  2 2 4 4 2 22
D_{4} 16  164  4 22 22 22 2 4 22 22
51. 0_{+} 6_{+} 3_{+}  -2 -19 20 1^{2} 3_{+} 2  2 1^{4} 3_{+} 41 41 2 22 4 2 2 4 2 22
D_{4} 8  18  1^{2} 3_{+} 2  2  2 22 22 22 4 2 22 22

Discriminant class: 41 (continued)

∞ 2 41 a_{1} a_{2} a_{3}  Over p = 2  Over p = 41  3 5 7 11 13 17 19 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 29 31 37 43 47 53 59 61
52. 2_{−} 6_{−} 3_{+}  2 -19 -20 1^{2} 3_{i} −2 −2 1^{4} 3_{+} 41 41 4 22 2 4 2 4 4 22
D_{4} 8  428  1^{2} 3_{i} −2 −2  2 22 22  2 2 22 22
53. 2_{−} 8_{−} 3_{+}  0 0 328 1^{4} 8_{−} 1  {−2 \atop −1} 1^{4} 3_{+} 41 41 2 22 4 2 2 4 2
D_{4} 64  1353  2  22 22 4 2 22 22
54. 2_{−} 8_{−} 3_{+}  0 82 -164 1^{4} 8_{−} 1  {−2 \atop −1} 1^{4} 3_{+} 41 41 4 22 2 4 2 4 4 22
D_{4} 32  82  2 22 22 22 2 2  22
55. 2_{−} 8_{−} 3_{+}  0 0  0 1^{4} 8_{−} 1 {−2∗ \atop −1} 1^{4} 3_{+} 41 41 2 22 2 2 4 4 2 22
D_{4} 32  41  4 22  22 2 4 22

Discriminant class: 82

∞  2 41 a_{1} a_{2} a_{3}  Over p = 2  Over p = 41 3 5 7 11 13 17 19 23
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
56. 0_{+} 11_{+} 3_{+}  0 -164 0 1^{4} 11_{+} 2  2 1^{4} 3_{+} 41 41  4 4 22  4 22 22
C_{4} 16  82  4 4 4 22 4 4
57. 2_{−} 11_{−} 3_{+}  0 164 0 1^{4} 11_{−} 2  2 1^{4} 3_{+} 41 41  4 4 22 22 4 22
C_{4} 160  6642  22 22 4 4 4  4 4
58. 2_{−} 11_{−} 3_{+}  0 164 0 1^{4} 11_{−} 2  2 1^{4} 3_{+} 41 41 22 4 4  4
C_{4} 128  82  22 4 4 4 22 4 4
59. 0_{+} 11_{+} 3_{+}  0 -164 0 1^{4} 11_{+} 2  2 1^{4} 3_{+} 41 41 22 4 4  22 4  22
C_{4} 16  6642  22  4 4 4  4 4
60. 2_{−} 9_{−} 1_{+}  0 -18 0 1^{4} 9_{−} 2 −1 1^{2} 1_{+} 41 41 22 2 4 22  2 22 22
D_{4}  4  82  22 22 2 4 4 22 4 2
61. 2_{−} 9_{−} 1_{+}  0 -2 0 1^{4} 9_{−} 2 −1 1^{2} 1_{+} 41 41 22 2 4 22 22 2 22 22
D_{4}  2  82  22 2 4 4  4 2
62. 2_{−} 11_{−} 1_{+}  0 20 0 1^{4} 11_{−} 2  2 1^{2} 1_{+} 41 41 22 4 2 22 22 2 22 22
D_{4}  6  82  22  4 4 2 22 4 4
63. 0_{+} 11_{+} 1_{+}  0 -20 0 1^{4} 11_{+} 2  2 1^{2} 1_{+} 41 41 22 4 2 22 22 2 22
D_{4}  2  82  22 22 4 4 2 22 4 4
64. 2_{−} 11_{−} 1_{+}  0 16 0 1^{4} 11_{−} 2 −2 1^{2} 1_{+} 41 41  4 4  22 2 22 22
D_{4}  2  82  22 22 4 2 4 22 2 4
65. 2_{−} 11_{−} 1_{+}  0 -16 0 1^{4} 11_{−} 2 −2 1^{2} 1_{+} 41 41 22 4 4 22 22 2  22
D_{4}  2  82  22 22 4 2 4 22 2 4
66. 0_{+} 3_{+} 3_{+}  -1 -15 23 1^{2} 3_{+} 2  2 1^{4} 3_{+} 41 41 22 2 4 22 22 4 22 22
D_{4}  2  -4  22 22 2 2 4 22 2 2
67. 2_{−} 5_{−} 3_{+}  0 41 0 1^{2} 3_{i} −2 −2 1^{4} 3_{+} 41 41 22 2 4 22 22 4 22
D_{4} 16  328  1^{2} 2_{i} −1 −1  22  2 2 4 22 2 2
68. 2_{−} 9_{−} 3_{+}  0 82 0 1^{4} 9_{−} 2 −1 1^{4} 3_{+} 41 41  2 2 22 22 4  22
D_{4} 32  2050  22 22 2 4 2 22 4 2
69. 2_{−} 9_{−} 3_{+}  0 -82 0 1^{4} 9_{−} 2 −1 1^{4} 3_{+} 41 41 22 2 2  22 4 22 22
D_{4} 32  2050  22 22 2 4 2 22 4 2
70. 2_{−} 11_{−} 3_{+}  0  0 0 1^{4} 11_{−} 2 −2 1^{4} 3_{+} 41 41 22 4 2 22 22 4 22 22
D_{4} 64  328  22 4 2 2 22 2 4
71. 2_{−} 11_{−} 3_{+}  0  0 0 1^{4} 11_{−} 2 −2 1^{4} 3_{+} 41 41 22 4 2 22  4 22 22
D_{4} 32  82  22 22 4 2 2  2 4