Quartic Fields Unramified Outside of {∞, 2, 47}

Discriminant class: −94

∞  2 47 a_{1} a_{2} a_{3}  Over p = 2  Over p = 47  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 41 43 53 59 61
1. 1_{k} 11_{−} 1_{k}  0 4 0 1^{4} 11_{−} 2 2 1^{2} 1_{k} 47 47 4 22 22 22 22 22 22 2
D_{4}  -94  2 0_{+} ∗  ∗ 22 2 4 2 22 4 4 4
2. 1_{k} 11_{+} 1_{i}  0 -4 0 1^{4} 11_{+} 2 2 1^{2} 1_{i} 47∗ 47∗ 4 22  22 22 22 22 2
D_{4}  2  -94  22 2 4 2 22 4 4 4
3. 1_{k} 9_{−} 3_{k}  0 94 0 1^{4} 9_{−} 2 −1 1^{4} 3_{k} 47∗ 47 4 22 22 22 22  22 2
D_{4} 12  -94  22 2 2 4  2 4 2
4. 1_{k} 9_{−} 3_{k}  0 -94 0 1^{4} 9_{−} 2 −1 1^{4} 3_{k} 47∗ 47 4 22 22  22  2
D_{4}  4  -94  22 2 2 4 22 2 4 2
5. 1_{k} 11_{−} 3_{k}  0 0 0 1^{4} 11_{−} 2 −2 1^{4} 3_{k} 47∗ 47 2 22 22 22  22 22 2
D_{4}  8  -94  2 4 4 22 4 2 4
6. 1_{k} 11_{−} 3_{k}  0 0 0 1^{4} 11_{−} 2 −2 1^{4} 3_{k} 47∗ 47 2  22 22 22 22 22 2
D_{4}  8  -376  22 2 4 4 22 4 2 4

Discriminant class: −47

∞ 2 47 a_{1} a_{2} a_{3}  Over p = 2  Over p = 47  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 41 43 53 59 61
7. 1_{k} 6_{+} 1_{i}  2 3 -4 1^{2} 3_{+} 2  2 1^{2} 1_{i} 47∗ 47∗ 22 4 22 4 4 22 4 2
D_{4}  -4  1^{2} 3_{+} 2  2  4 2 22 2 4 22 22 22
8. 1_{k} 8_{−} 1_{k}  0 -6 12 1^{4} 8_{−} 1 {−2 \atop −1} 1^{2} 1_{k} 47 47 22 4  4 4 22 4 2
D_{4}  -14  2 0_{+} ∗  ∗ 4 2 22 2 4 22 22 22
9. 1_{k} 4_{−} 3_{k}  0 0  0 1^{2} 2_{i} −1 −1 1^{4} 3_{k} 47∗ 47 22 4 22 2 4  2 2
D_{4}  -188  1^{2} 2_{i} −1 −1  4 2 22 4 2 22 22 22
10. 1_{k} 6_{−} 3_{k}  -2 25 -24 1^{2} 3_{i} −2 −2 1^{4} 3_{k} 47∗ 47 22 2 22 4 2 22 4 2
D_{4}  -232  1^{2} 3_{i} −2 −2  2 2 22 4 4 22  22
11. 1_{k} 8_{−} 3_{k}  0 -94 -564 1^{4} 8_{−} 1 {−2 \atop −1} 1^{4} 3_{k} 47∗ 47  2 22 4 2 22 4 2
D_{4} 4  -846  2 2 22 4 4 22 22 22
12. 1_{k} 8_{−} 3_{k}  0 0  0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{4} 3_{k} 47∗ 47 22 4 22 2 4  2 2
D_{4} 8  -47  4 2  4 2  22

Discriminant class: −2

∞  2 47  a_{1} a_{2} a_{3}  Over p = 2  Over p = 47  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 41 43 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 22 2
D_{4}  -2  4 2 4 22 22 4 22 4
14. 1_{k} 11_{i} 2_{+}  0 0 0 1^{4} 11_{i} −2  2 1^{2} 1_{k} 47 47 22 4 2 22 4 22 22 2
D_{4}  4  -4418  1^{2} 1_{i} 47∗ 47∗ 4 2 4  22 4 22 4
15. 1_{k} 9_{i} 2_{+}  0 -16 -4 1^{4} 9_{i} −2 −∗ 1^{3} 2_{+} ∗  3 4 4 22 4 3  2
S_{4}  57  2 4 2 3 3 4 3 4
16. 1_{k} 11_{i} 2_{+}  0 -24 -64 1^{4} 11_{i} −2  2 1^{3} 2_{+} ∗  3 4 4 22 2 3 22 2
S_{4}  2  46  2 2 4 3 3 4 3 2

Discriminant class: −2 (continued)

∞  2 47  a_{1} a_{2} a_{3}  Over p = 2  Over p = 47  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 41 43 53 59 61
17. 1_{k} 11_{i} 2_{+}  0 -16 184 1^{4} 11_{i} −2 −2∗ 1^{3} 2_{+} ∗  3 2 2  4 3 22 2
S_{4}  10  2 4 4 3 3 2 3 4

Discriminant class: −1

∞  2 47  a_{1} a_{2} a_{3}  Over p = 2  Over p = 47  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 41 43 53 59 61
18. 1_{k} 10_{i} 0_{+}  0 -2 0 1^{4} 10_{i} −1 2 2 0_{+} ∗  ∗ 4 22 2 4 22 22 4 2
D_{4}  -1  22 2 22  4 22 4 22
19. 1_{k} 10_{i} 2_{+}  0 -94 0 1^{4} 10_{i} −1 2 1^{2} 1_{i} 47∗ 47∗ 4 22 2 4 22 22 4 2
D_{4}  4  -2209  1^{2} 1_{k} 47 47 22 2 22 22 4 22 4 22
20. 1_{k} 6_{i} 2_{+}  -2 7 0 1^{2} 3_{i} −2 −2 1^{3} 2_{+} ∗  4 3 4 4 3 3 2 4
S_{4}  -8  1^{2} 3_{+} 2  2  4 22  4 3 2 3
21. 1_{k} 10_{i} 2_{+}  0 -6 8 1^{4} 10_{i} −1 2 1^{3} 2_{+} ∗  4 3 4 2 3 3 2 2
S_{4}  4  -97  22 2 22 22 4 3 4 3
22. 1_{k} 10_{i} 2_{+}  0 -14 -16 1^{4} 10_{i} −1 2 1^{3} 2_{+} ∗  2 3 2 4 3 3 2 4
S_{4}  4  15  22 4  22 2 3 4 3

Discriminant class: 1

∞ 2 47 a_{1} a_{2} a_{3}  Over p = 2  Over p = 47  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 41 43 53 59 61
23. 2_{−} 8_{−} 0_{+}  0 0 0 1^{4} 8_{−} 1 {−2 \atop −1} 2 0_{+} ∗  ∗ 22 22 22 22 22  22 22
V_{4}  1  2 0_{+} ∗  ∗ 22 22 22  22 22 22 22
24. 2_{−} 4_{−} 2_{+}  0 -23 0 1^{2} 2_{i} −1 −1 2^{2} 2_{+} 1  V 22 22 22 22 22  22 22
V_{4} 5  144  1^{2} 2_{i} −1 −1  22 22  22 22  22
25. 2_{−} 6_{−} 2_{+}  2 29 28 1^{2} 3_{i} −2 −2 2^{2} 2_{+} 1  V  22 22 22 22  22 22
V_{4} 5  102  1^{2} 3_{i} −2 −2  22 22 22 22 22 22  22
26. 2_{−} 6_{+} 2_{−}  2 21 20 1^{2} 3_{+} 2  2 1^{2} 1_{i} 47∗ 47∗ 22 22  22 22  22 22
V_{4} 20  194  1^{2} 3_{+} 2  2 1^{2} 1_{i} 47∗ 47∗ 22 22 22 22 22 22 22 22
27. 2_{−} 8_{−} 2_{+}  0 0 0 1^{4} 8_{−} 1 {−2 \atop −1} 2^{2} 2_{+} 1  V 22  22 22  22 22
V_{4} 8  2209  22 22 22 22 22 22 22
28. 2_{−} 8_{−} 2_{+}  0 48 0 1^{4} 8_{−} 1 {−2 \atop −1} 2^{2} 2_{+} 1  V 22 22 22  22  22
V_{4} 8  529  22 22 22 22  22 22 22
29. 0_{+} 8_{−} 2_{−}  0 -48 0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{2} 1_{k} 47 47 22 22 22 22 22  22
V_{4}  529  1^{2} 1_{k} 47 47 22  22 22 22 22 22 22

Discriminant class: 2

∞  2 47 a_{1} a_{2} a_{3}  Over p = 2  Over p = 47  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 41 43 53 59 61
30. 2_{−} 11_{−} 0_{+}  0  4 0 1^{4} 11_{−} 2  2 2 0_{+} ∗  ∗ 4 4  4 4  4
C_{4}  2  2 0_{+} ∗  ∗ 4 22 4 22 4 4 4 4

Discriminant class: 2 (continued)

∞  2 47 a_{1} a_{2} a_{3}  Over p = 2  Over p = 47  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 41 43 53 59 61
31. 0_{+} 11_{+} 0_{+}  0 -4 0 1^{4} 11_{+} 2  2  4 4 22 4 4  4 22
C_{4}  2  4  4 22 4 4 4 4
32. 0_{+} 11_{−} 2_{−}  0 -188 0 1^{4} 11_{−} 2  2 1^{2} 1_{k} 47 47 4 4  4 4  4 22
C_{4}  4418  1^{2} 1_{k} 47 47 4  4  4 4 4 4
33. 2_{−} 11_{+} 2_{−}  0 188 0 1^{4} 11_{+} 2  2 1^{2} 1_{i} 47∗ 47∗ 4 4 22 4 4  4
C_{4} 32  4418  1^{2} 1_{i} 47∗ 47∗ 4 22 4  4 4 4 4
34. 2_{−} 9_{−} 0_{+}  0 -2 0 1^{4} 9_{−} 2 −1 2 0_{+} ∗  ∗ 4 2 22 4 2 22 4 22
D_{4}  2  2 0_{+} ∗  ∗ 2 22 2  4 2 4 2
35. 2_{−} 11_{−} 0_{+}  0  0 0 1^{4} 11_{−} 2 −2 2 0_{+} ∗  ∗ 2 4 22 2 4 22 2 22
D_{4}  2  2 0_{+} ∗  ∗ 4 22 4 22 2 4 2 4
36. 2_{−} 3_{+} 2_{−}  0  5 0 1^{2} 3_{+} 2  2 1^{2} 1_{i} 47∗ 47∗ 2 4 22 4 4 22 4 22
D_{4}  5  18  1^{2} 1_{i} 47∗ 47∗ 4 22 2 22 4 2 2 2
37. 2_{−} 5_{−} 2_{+}  0 -5 0 1^{2} 3_{i} −2 −2 2^{2} 2_{+} 1  V 2 4  4 4 22 4 22
D_{4}  5  18  1^{2} 2_{i} −1 −1  4 22 2 22 4 2 2 2
38. 2_{−} 9_{−} 2_{+}  0 14 0 1^{4} 9_{−} 2 −1 2^{2} 2_{+} 1  V 4 4 22 2 4  2 22
D_{4}  4  2  4  2 22 2 2 4 2
39. 0_{+} 9_{−} 2_{−}  0 -14 0 1^{4} 9_{−} 2 −1 1^{2} 1_{k} 47 47 4 4 22 2 4  2
D_{4}  2  1^{2} 1_{k} 47 47 4 22 2 22 2 2 4 2
40. 2_{−} 9_{−} 2_{+}  0 94 0 1^{4} 9_{−} 2 −1 2^{2} 2_{+} 1  V 4 2 22 4 2 22 4 22
D_{4}  8  4418  2 22 2 22 4 2 4 2
41. 0_{+} 11_{−} 2_{−}  0 -24 0 1^{4} 11_{−} 2 −2 1^{2} 1_{k} 47 47 2 2 22 4 2  4 22
D_{4}  50  1^{2} 1_{k} 47 47 2 22 4 22 4 4 2 4
42. 2_{−} 11_{+} 2_{−}  0 -4 0 1^{4} 11_{+} 2  2 1^{2} 1_{i} 47∗ 47∗ 4 2  2 2 22 2 22
D_{4} 16  98  1^{2} 1_{i} 47∗ 47∗ 2 22 4 22 2 4 4 4
43. 2_{−} 11_{−} 2_{+}  0  4 0 1^{4} 11_{−} 2  2 2^{2} 2_{+} 1  V 4 2 22 2 2 22 2 22
D_{4}  8  98  2 22 4 22 2 4 4 4
44. 2_{−} 11_{−} 2_{+}  0 24 0 1^{4} 11_{−} 2 −2 2^{2} 2_{+} 1  V 2 2 22 4 2  4
D_{4}  8  50  2  4 22 4 4 2 4
45. 2_{−} 11_{−} 2_{+}  0  0 0 1^{4} 11_{−} 2 −2 2^{2} 2_{+} 1  V 2 4 22 2 4 22 2 22
D_{4}  4  4418  4 22 4  2 4 2 4

Discriminant class: 47

∞  2 47 a_{1} a_{2} a_{3}  Over p = 2  Over p = 47  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 41 43 53 59 61
46. 0_{+} 10_{i} 1_{k}  0 -14 0 1^{4} 10_{i} −1 2 1^{2} 1_{k} 47 47 4 4 2 22 4  22
D_{4}  47  4 22 22 2 22 22 4 22
47. 2_{−} 10_{i} 1_{i}  0 14 0 1^{4} 10_{i} −1 2 1^{2} 1_{i} 47∗ 47∗ 4 4 2 22 4  22 22
D_{4}  4  47  2 0_{+} ∗  ∗ 4  22 2 22 22 4 22
48. 2_{−} 2_{i} 3_{i}  0 0 0 1^{2} 2_{i} −1 −1 1^{4} 3_{i} 47 47∗ 2 4 2 22 4  22 22
D_{4}  5  47  4 22 22 4 22 22 2 22
49. 2_{−} 6_{i} 3_{i}  0 0 0 1^{2} 3_{i} −2 −2 1^{4} 3_{i} 47 47∗ 2 4 2 22 4  22 22
D_{4} 40  188  1^{2} 3_{+} 2  2  4 22  4 22  2
50. 2_{−} 10_{i} 3_{i}  0 94 0 1^{4} 10_{i} −1 2 1^{4} 3_{i} 47 47∗ 4 2 2  2  22
D_{4} 32  2303  2 22 22 4 22 22 4 22
51. 2_{−} 10_{i} 3_{i}  0 -94 0 1^{4} 10_{i} −1 2 1^{4} 3_{i} 47 47∗ 4 2 2 22 2  22 22
D_{4} 96  2303  2 22 22 4  22 4 22

Discriminant class: 94

∞  2 47 a_{1} a_{2} a_{3}  Over p = 2  Over p = 47  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 41 43 53 59 61
52. 0_{+} 11_{i} 1_{k}  0 -24 0 1^{4} 11_{i} −2 2 1^{2} 1_{k} 47 47 22 22 2 4 22  4 22
D_{4}  94  22 22 4 2 4 4 22 4
53. 2_{−} 11_{i} 1_{i}  0 24 0 1^{4} 11_{i} −2 2 1^{2} 1_{i} 47∗ 47∗ 22 22 2 4 22  4
D_{4}  8  94  2 0_{+} ∗  ∗ 22  4 2 4 4 22 4
54. 2_{−} 3_{i} 3_{i}  1 18 -44 1^{2} 3_{i} −2 −2 1^{4} 3_{i} 47 47∗ 22 22 2 4 22 22 4 22
D_{4}  5  56  22 22 2 4 4 2  2
55. 2_{−} 5_{i} 3_{i}  0 47 0 1^{2} 3_{+} 2  2 1^{4} 3_{i} 47 47∗  22 2 4 22 22 4 22
D_{4} 20  846  1^{2} 2_{i} −1 −1  22 22 2 4 4 2 22 2
56. 2_{−} 11_{i} 3_{i}  0 0 0 1^{4} 11_{i} −2 2 1^{4} 3_{i} 47 47∗ 22  2 2 22 22 2 22
D_{4} 64  94  22 22 4 4 2 4 22 4
57. 2_{−} 11_{i} 3_{i}  0 0 0 1^{4} 11_{i} −2 2 1^{4} 3_{i} 47 47∗ 22 22 2 2  22 2 22
D_{4} 64  376  22 4 4 2 4 22 4