Quartic Fields Ramifying Above {2, 3}

Discriminant class: −6

∞  2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3 5 7 11 13 17 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_{−} 3_{k} 0 0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{k} 3 3∗ 22 22 22 4 4 2 2
D_{4} -24 22 4 4 2 2 22 4
2. 1_{k} 11_{−} 3_{k} 0 0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{k} 3 3∗  22 22 4 4 2 2 22
D_{4} -6 22 4 4 2 2 22 22 4
3. 1_{k} 9_{−} 3_{k} 0 0 -4 1^{4} 9_{−} 2∗ −1 1^{3} 3_{k} 3  3 3 3 4 4 4 4 22
S_{4} -3 3 4 4 2 2 3 22 2
4. 1_{k} 11_{−} 3_{k} 0 -12 -16 1^{4} 11_{−} 2∗ −2∗ 1^{3} 3_{k} 3  3 3 3 2 2 4 4 22
S_{4} 12 3 4 4 4 4 3 22 2
5. 1_{k} 11_{−} 3_{k} 0 0 -8 1^{4} 11_{−} 2∗ 2∗ 1^{3} 3_{k} 3  3 3 3 4 4 2 2 22
S_{4} 4 6 3 2 2 4 4 3 22 2

Discriminant class: −3

∞ 2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3 5 7 11 13 17 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
6. 1_{k} 6_{+} 3_{i} 2 0 2 1^{4} 6_{+} ∗ −∗ 1^{4} 3_{i} 3∗ 3 4 22 2 22 4 22 2 4
D_{4} 1 22 4 22 2 4 2
7. 1_{k} 8_{+} 3_{i} 0 0 0 1^{4} 8_{+} ∗ −∗ 1^{4} 3_{i} 3∗ 3 4 22 2  4 22 2 4
D_{4}  -3  22 22 4 22 2 4 2 22
8. 1_{k} 4_{+} 5_{i} 2 0 6 1^{4} 4_{+} ∗  1^{3} 5_{i} 3∗  4 22 4 3 4 3 4 2
S_{4}  3  3 22 2 3 2 4 4 3
9. 1_{k} 8_{+} 3_{i} 0 0 4 1^{4} 8_{+} ∗  1^{3} 3_{i} 3∗  4 3 4 3 4 3 4 2
S_{4}  -6  22 3 2 22 4 2 4 3
10. 1_{k} 6_{+} 5_{i} -2 0 -4 1^{4} 6_{+} ∗ −1 1^{3} 5_{i} 3∗  4 3 4 3 2 3 2 4
S_{4} 2 3 3 4 3 4 4 4 22
11. 1_{k} 6_{+} 5_{i} 2 -3 2 2^{2} 6_{+} ∗ ∗ 1^{3} 5_{i} 3∗  4 3 2 3 4 3 4 2
S_{4} 2 7 3 3 4 3 2 2 4 22
12. 1_{k} 8_{+} 5_{i} 0 -6 -4 1^{4} 8_{+} ∗ −∗ 1^{3} 5_{i} 3∗  2 3 4 3 4 3 4 4
S_{4}  6  3 3 2 3 4 4 2 22
13. 1_{k} 8_{+} 5_{i} 0 -6 -4 1^{4} 8_{+} ∗  1^{3} 5_{i} 3∗  2 3 4 22 2 22 4 4
S_{4}  15  3 3 4 3 2 2 4 3

Discriminant class: −2

∞  2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  5 7 11 13 17 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
14. 1_{k} 9_{k} 2_{−} 0 -2 0 1^{4} 9_{k} −2 −∗ 1^{2} 1_{k} 3 3 4 4 22 2 22 22 2 4
D_{4} -2 1^{2} 1_{k} 3 3 4 2 22 22 2 4 2
15. 1_{k} 9_{i} 2_{+} 0 2 0 1^{4} 9_{i} −2 −∗ 2^{2} 2_{+} 1 V 4 4  2 22 22 2 4
D_{4} -2 4 2 22 22 2 4 22 2
16. 1_{k} 11_{i} 2_{+} 0 -4 0 1^{4} 11_{i} −2 −2∗ 2^{2} 2_{+} 1 V 2 4 22 4 22  2 2
D_{4}  -2  4 4 22  2 2 22 4

Discriminant class: −2 (continued)

∞  2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  5 7 11 13 17 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_{k} 2_{−} 0 4 0 1^{4} 11_{k} −2 −2∗ 1^{2} 1_{i} 3∗ 3∗ 2 4 22 4 22 22 2 2
D_{4}  -2  1^{2} 1_{i} 3∗ 3∗ 4 4 22 22 2 2 22 4
18. 1_{k} 11_{i} 2_{+} 0 0 0 1^{4} 11_{i} −2  2 2^{2} 2_{+} 1 V 4 2 22 4  22 2 4
D_{4}  -18  2 4  22 2 4 22 4
19. 1_{k} 9_{i} 4_{+} 0 -6 -8 1^{4} 9_{i} −2 −∗ 1^{3} 4_{+} 1  4 2 3 4 3 3 4 2
S_{4}  6  2 4 3 3 4 2 3 2
20. 1_{k} 11_{i} 4_{+} 0 -12 -8 1^{4} 11_{i} −2  2 1^{3} 4_{+} 1  2 4 3 4 3 3 4 4
S_{4}  2  18  2 2 3 3 4 2 3 2
21. 1_{k} 11_{i} 4_{+} 0 0 -8 1^{4} 11_{i} −2 −2∗ 1^{3} 4_{+} 1  4 4 3 2 3 3 2 4
S_{4}  -6  2 4 3 3 2 2 3 2

Discriminant class: −1

∞  2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3 5 7 11 13 17 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
22. 1_{k} 10_{k} 2_{−} 0 6 0 1^{4} 10_{k} −1 2 2^{2} 2_{−} ∗ ∗ 22 2 4 22  4 2 22
D_{4} -9 2 22 22 4 2 22 4 22
23. 1_{k} 6_{i} 4_{+} 2 3 -2 2^{2} 6_{i} −1 ∗ 1^{3} 4_{+} ∗  3 4 4 3 3 4 4 3
S_{4} -1 2 3 22 4 2 22 2 3
24. 1_{k} 10_{i} 4_{+} 0 0 -16 1^{4} 10_{i} −1 2∗ 1^{3} 4_{+} ∗  3 2 4 3 3 2 4 3
S_{4} 2 -24 4 3 22 4 4 22 2 3
25. 1_{k} 10_{i} 4_{+} 0 -6 -8 1^{4} 10_{i} −1 2 1^{3} 4_{+} ∗  3 4 2 3 3 4 2 3
S_{4}  2  15  4 3 22 2 4  2 3

Discriminant class: 1

∞ 2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  5 7 11 13 17 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
26. 2_{−} 4_{−} 2_{+} 0 -1 0 2^{2} 4_{−} 1 {−1 \atop −∗} 2^{2} 2_{+} 1 V 22 22 22  22 22 22 22
V_{4}  1  22  22 22 22 22 22
27. 2_{−} 6_{−} 2_{+} 0 2 0 2^{2} 6_{−} 1  {∗ \atop 2} 2^{2} 2_{+} 1 V 22  22 22 22 22 22 22
V_{4}  4  22 22 22 22 22 22 22
28. 2_{−} 6_{+} 2_{−} 0 -2 0 2^{2} 6_{+} 1 {−2 \atop −2∗} 1^{2} 1_{i} 3∗ 3∗ 22 22 22 22 22  22 22
V_{4} 4 1^{2} 1_{i} 3∗ 3∗ 22 22 22 22 22 22 22
29. 2_{−} 8_{+} 2_{−} 0 4 0 1^{4} 8_{+} 1 {−2 \atop −∗} 1^{2} 1_{k} 3 3 22 22  22 22 22 22 22
V_{4} 2 1 1^{2} 1_{k} 3 3 22 22 22 22 22 22 22
30. 0_{+} 8_{+} 2_{+} 0 -4 0 1^{4} 8_{+} 1 {−2∗ \atop −∗} 2^{2} 2_{+} 1 V 22 22 22 22 22 22  22
V_{4} 1 22 22 22 22 22 22 22
31. 2_{−} 8_{−} 2_{+} 0 0 0 1^{4} 8_{−} 1 {−2∗ \atop −1} 2^{2} 2_{+} 1 V  22 22 22 22 22 22
V_{4} 2  9  22 22 22 22 22  22 22
32. 2_{−} 6_{−} 4_{+} -2 6 -4 1^{4} 6_{−} 1  1^{3} 4_{+} 1  3 3 3 3 22 22 3 3
A_{4}  2  3 22 3 3 3 22 3 3

Discriminant class: 2

∞  2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  5 7 11 13 17 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
33. 2_{−} 11_{+} 2_{−} 0 12 0 1^{4} 11_{+} 2 2 2^{2} 2_{−} ∗ ∗ 4 22 4 4 22 4  4
C_{4} 2 18 4 4 22 4 4 4
34. 0_{+} 11_{−} 2_{−} 0 -12 0 1^{4} 11_{−} 2 2 2^{2} 2_{−} ∗ ∗ 4  4 4 22 4 22 4
C_{4} 18 22 4 4 4 4 4
35. 2_{−} 9_{+} 2_{−} 0 6 0 1^{4} 9_{+} 2 −1 2^{2} 2_{−} ∗ ∗ 2 22 4 2  4 22 2
D_{4} 2 18 22 2 22 4 22 2 4 2
36. 2_{−} 11_{−} 2_{+} 0 0 0 1^{4} 11_{−} 2 −2 1^{2} 1_{i} 3∗ 3∗ 4 22 2 4  2 22 4
D_{4}  2  18  1^{2} 1_{k} 3 3 22 4  2 22 4 2 4

Discriminant class: 3

∞  2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  5 7 11 13 17 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
37. 2_{−} 4_{k} 3_{k} 0 -3 0 2^{2} 4_{k} −∗  ∗ 1^{4} 3_{k} 3 3∗ 4 2 22 22 4 2 22 4
D_{4}  3  2  4 2 22 4 22
38. 2_{−} 6_{k} 3_{k} 0 0 0 2^{2} 6_{k} −∗  ∗ 1^{4} 3_{k} 3 3∗ 4 2 22  4 2 22 4
D_{4}  12  2 22 4 2 22 4 22 22
39. 2_{−} 10_{k} 1_{k} 0 2 0 1^{4} 10_{k} −∗ −2 1^{2} 1_{k} 3 3 4 4 22 22 2 2 22 4
D_{4} 3 4 22 2 2 22 4 22
40. 2_{−} 10_{i} 1_{i} 0 -2 0 1^{4} 10_{i} −∗ −2 1^{2} 1_{i} 3∗ 3∗ 4 4  22 2 2 22 4
D_{4} 3 2 0_{+} ∗ ∗ 4 22 2 2 22 4 22 22
41. 2_{−} 10_{k} 3_{k} 0 6 0 1^{4} 10_{k} −∗ −2∗ 1^{4} 3_{k} 3 3∗ 2 4 22 22 4 2  2
D_{4} 2 3 4 22 4 2 2 22 22
42. 0_{+} 10_{i} 3_{k} 0 -6 0 1^{4} 10_{i} −∗ −2∗ 1^{4} 3_{k} 3 3∗ 2 4 22 22 4 2 22 2
D_{4}  3  4 22 4 2 22 2 22 22

Discriminant class: 6

∞  2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  5 7 11 13 17 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
43. 2_{−} 11_{k} 1_{k} 0 4 0 1^{4} 11_{k} −2∗ −2 1^{2} 1_{k} 3 3 22 4 2 4 2  22 22
D_{4} 6 2 0_{+} ∗ ∗ 4 4 2 22 22 2 4
44. 2_{−} 11_{i} 1_{i}  0 -4 0 1^{4} 11_{i} −2∗ −2 1^{2} 1_{i} 3∗ 3∗ 22 4 2 4 2 22 22 22
D_{4}  6  4 4 2 22 22 22 2 4
45. 2_{−} 9_{i} 3_{i}  0 6 0 1^{4} 9_{i} −2∗ −∗ 1^{4} 3_{i} 3∗ 3 22 4 2 2 4 22  22
D_{4}  2  6  4 2 4 22  22 2 2
46. 0_{+} 9_{k} 3_{i}  0 -6 0 1^{4} 9_{k} −2∗ −∗ 1^{4} 3_{i} 3∗ 3 22 4 2 2 4 22 22 22
D_{4} 6 4 2 4 22 22 22 2 2
47. 2_{−} 11_{i} 3_{i}  0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{i} 3∗ 3 22 2 2 4 4 22 22
D_{4}  2  6  2 4 4 22 22  2 4
48. 2_{−} 11_{i} 3_{i}  0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{i} 3∗ 3  2 2 4 4 22 22 22
D_{4}  2  24  2 4 4 22 22 22 2 4
49. 2_{−} 5_{i} 5_{i}  0 3 -2 1^{2} 3_{+} 2  2 1^{3} 5_{i} 3∗  3 4 4 4 4 3 3 3
S_{4}  6  1^{2} 2_{i} −∗ −∗  2 2 4 22 3 3 4 4
50. 2_{−} 9_{i} 5_{i}  0 12 -16 1^{4} 9_{i} −2∗ −∗ 1^{3} 5_{i} 3∗  3 2 2 4 2 3 3 3
S_{4}  4  24  4 2 4 22 3 3 4 2

Discriminant class: 6 (continued)

∞  2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  5 7 11 13 17 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
51. 2_{−} 9_{i} 5_{i}  0 12 -4 1^{4} 9_{i} −2∗ −∗ 1^{3} 5_{i} 3∗  3 4 4 2 4 3 3 3
S_{4}  4  69  4 2 2  3 3 2 4
52. 2_{−} 11_{i} 5_{i}  0 12 -16 1^{4} 11_{i} −2∗ 2∗ 1^{3} 5_{i} 3∗  3 4 2 2 4 3 3 3
S_{4}  2  60  4 2 4 22 3 3 2 4
53. 2_{−} 11_{i} 5_{i}  0 12 -8 1^{4} 11_{i} −2∗ 2∗ 1^{3} 5_{i} 3∗  3 2 4 4 2 3 3 3
S_{4}  2  42  4 2 2 22 3 3 4 2
54. 0_{+} 11_{k} 5_{i}  0 -24 56 1^{4} 11_{k} −2∗ −2 1^{3} 5_{i} 3∗  3 4 2 4 4 3 3 3
S_{4}  -30  2 2 2 22 3 3 4 4
55. 2_{−} 11_{i} 5_{i}  0 12 -16 1^{4} 11_{i} −2∗ −2 1^{3} 5_{i} 3∗  3 2 4 2 2 3 3 3
S_{4}  2  6  2 2 4 22 3 3 2 2