Quartic Fields Ramifying Above {2, 31}

Discriminant class: −62

∞  2 31 a_{1} a_{2} a_{3}  Over p = 2  Over p = 31  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 37 41 43 47 53 59 61
1. 1_{k} 11_{+} 1_{i}  0 12 0 1^{4} 11_{+} 2 2 1^{2} 1_{i} 31∗ 31∗ 22 4  22 22 2 4 2
D_{4}  2  -62  22 22  22 22 22 4 22
2. 1_{k} 11_{−} 1_{k}  0 -12 0 1^{4} 11_{−} 2 2 1^{2} 1_{k} 31 31 22 4 22 22 22 2 4 2
D_{4}  -62  2 0_{+} ∗  ∗ 22 22  22  22 4 22
3. 1_{k} 9_{−} 3_{k}  0 62 0 1^{4} 9_{−} 2 −1 1^{4} 3_{k} 31∗ 31  2 22  22 4 4 2
D_{4}  2  -558  22 22 22 22 22 22 4 22
4. 1_{k} 9_{−} 3_{k}  0 -62 0 1^{4} 9_{−} 2 −1 1^{4} 3_{k} 31∗ 31 22 2 22 22 22 4 4 2
D_{4} 2 -558 22 22 22 22 22 4 22
5. 1_{k} 11_{−} 3_{k}  0 0 0 1^{4} 11_{−} 2 −2 1^{4} 3_{k} 31∗ 31 22 4 22 22 22 4 2 2
D_{4}  2  -62  22 22 22 22 22 22 2
6. 1_{k} 11_{−} 3_{k}  0 0 0 1^{4} 11_{−} 2 −2 1^{4} 3_{k} 31∗ 31 22 4 22 22  4 2 2
D_{4} 10  -248  22 22 22  2 22

Discriminant class: −31

∞ 2 31 a_{1} a_{2} a_{3}  Over p = 2  Over p = 31  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 37 41 43 47 53 59 61
7. 1_{k} 6_{+} 1_{i}  2 1  0 1^{2} 3_{+} 2  2 1^{2} 1_{i} 31∗ 31∗ 4 22 22 4 4 2 22 2
D_{4}  -2  1^{2} 3_{+} 2  2  4 4 22 4 22 4 22 4
8. 1_{k} 8_{−} 1_{k}  0 0 -8 1^{4} 8_{−} 1 {−2 \atop −1} 1^{2} 1_{k} 31 31 4 22  4 4 2 22 2
D_{4}  -7  2 0_{+} ∗  ∗ 4 4 22 4  4 22 4
9. 1_{k} 4_{−} 3_{k}  2 -14 16 1^{2} 2_{i} −1 −1 1^{4} 3_{k} 31∗ 31 2 22 22 2 4 4 22 2
D_{4}  64  1^{2} 2_{i} −1 −1  4 4  2 22 4 22 4
10. 1_{k} 6_{−} 3_{k}  -2 17 -16 1^{2} 3_{i} −2 −2 1^{4} 3_{k} 31∗ 31 4 22 22 4 2 4  2
D_{4}  2  1^{2} 3_{i} −2 −2  2 2 22 4 22 2 22 2
11. 1_{k} 8_{−} 3_{k}  0 0  0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{4} 3_{k} 31∗ 31 2  22 2 4 4 22 2
D_{4} 4  -31  4 4  2 22 4 22 4
12. 1_{k} 8_{−} 3_{k}  0 0 -248 1^{4} 8_{−} 1 {−2 \atop −1} 1^{4} 3_{k} 31∗ 31 4 22 22 4 2 4 22 2
D_{4} 4  713  2 2 22 4 22 2  2
13. 1_{k} 6_{−} 1_{k}  2 2 -2 1^{4} 6_{−} 1  1^{2} 1_{k} 31 31 4 3 3 2 4 4 3 4
S_{4}  -1  2 0_{+} ∗  ∗ 4 2 3 4 22 2 3 4
14. 1_{k} 6_{−} 3_{k}  -2 -14 46 1^{4} 6_{−} 1  1^{4} 3_{k} 31∗ 31 2 3 3 4 2 2 3 2
S_{4}  -153  4 4 3 4  2 3 2
15. 1_{k} 6_{−} 3_{k}  2 -14 -46 1^{4} 6_{−} 1  1^{4} 3_{k} 31∗ 31 4 3 3 4 4 4 3 4
S_{4}  -29  2 4 3 2 22 2 3 4

Discriminant class: −2

∞  2 31  a_{1} a_{2} a_{3}  Over p = 2  Over p = 31  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 37 41 43 47 53 59 61
16. 1_{k} 11_{i} 2_{+}  0 0 0 1^{4} 11_{i} −2 2 1^{2} 1_{k} 31 31 22 4 2 22 4  22 2
D_{4}  8  -1922  1^{2} 1_{i} 31∗ 31∗ 4 4 22 22 2 4 22 4

Discriminant class: −1

∞  2 31 a_{1} a_{2} a_{3}  Over p = 2  Over p = 31  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 37 41 43 47 53 59 61
17. 1_{k} 10_{i} 2_{+}  0 -62 0 1^{4} 10_{i} −1 2 1^{2} 1_{i} 31∗ 31∗ 4 22 2 4 22  4 2
D_{4}  4  -961  1^{2} 1_{k} 31 31 22 22  4 2 22 4 22

Discriminant class: 1

∞ 2 31 a_{1} a_{2} a_{3}  Over p = 2  Over p = 31  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 37 41 43 47 53 59 61
18. 2_{−} 4_{−} 2_{+} 0 -15 0 1^{2} 2_{i} −1 −1 2^{2} 2_{+} 1  V 22  22 22 22 22 22 22
V_{4} 3  64  1^{2} 2_{i} −1 −1  22 22  22 22 22 22 22
19. 2_{−} 6_{−} 2_{+} 2 21 20 1^{2} 3_{i} −2 −2 2^{2} 2_{+} 1  V 22 22 22 22 22 22  22
V_{4} 3  38  1^{2} 3_{i} −2 −2  22 22  22 22 22  22
20. 2_{−} 6_{+} 2_{−} -2 13 -12 1^{2} 3_{+} 2  2 1^{2} 1_{i} 31∗ 31∗ 22 22  22 22 22 22 22
V_{4} 12  98  1^{2} 3_{+} 2  2 1^{2} 1_{i} 31∗ 31∗ 22 22  22  22 22 22
21. 2_{−} 8_{−} 2_{+} 0 32 0 1^{4} 8_{−} 1 {−2 \atop −1} 2^{2} 2_{+} 1  V  22 22  22 22 22 22
V_{4} 8  225  22 22  22 22 22 22
22. 0_{+} 8_{−} 2_{−} 0 -32 0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{2} 1_{k} 31 31 22 22 22 22 22 22 22
V_{4}  225  1^{2} 1_{k} 31 31 22 22  22 22 22 22 22
23. 2_{−} 8_{−} 2_{+} 0 0 0 1^{4} 8_{−} 1 {−2 \atop −1} 2^{2} 2_{+} 1  V 22 22 22 22  22 22 22
V_{4} 8  961  22 22  22
24. 2_{−} 4_{−} 2_{+} 0 7 -2 1^{2} 2_{i} −∗ −∗ 1^{3} 2_{+} 1  3 3 3 3 3 3 3 22
A_{4} 4 14 1^{2} 2_{i} −∗ −∗ 22 3 3 3 22 3 3 22
25. 0_{+} 6_{+} 2_{+} -2 -7 6 2^{2} 6_{+} 1 {−2 \atop −2∗} 1^{3} 2_{+} 1  3 3 3 3 3 3 3 22
A_{4}  11  22 3 3 3 22 3 3 22
26. 2_{−} 6_{−} 2_{+} 0 -6 -8 2^{2} 6_{−} 1  {∗ \atop 2} 1^{3} 2_{+} 1  3 3 3 3 3 3 3 22
A_{4} 2  28  22 3 3 3 22 3 3
27. 2_{−} 8_{−} 2_{+} 0 2 -4 1^{4} 8_{−} 1 {−2∗ \atop −1} 1^{3} 2_{+} 1  3 3 3 3 3 3 3 22
A_{4} 4 10 22 3 3 3 3 3 22
28. 2_{−} 8_{−} 2_{+} 0 10 -4 1^{4} 8_{−} 1 {−2 \atop −1} 1^{3} 2_{+} 1  3 3 3 3 3 3 3
A_{4} 4  2  3 3 3 22 3 3 22

Discriminant class: 2

∞  2 31 a_{1} a_{2} a_{3}  Over p = 2  Over p = 31  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 37 41 43 47 53 59 61
29. 2_{−} 11_{+} 2_{−}  0 124 0 1^{4} 11_{+} 2  2 1^{2} 1_{i} 31∗ 31∗ 4 4 22 4 4 22 4
C_{4} 16  1922  1^{2} 1_{i} 31∗ 31∗ 4 4 22 4  4 4 4
30. 0_{+} 11_{−} 2_{−}  0 -124 0 1^{4} 11_{−} 2  2 1^{2} 1_{k} 31 31 4 4  4 4 22 4 22
C_{4}  1922  1^{2} 1_{k} 31 31 4 4 22 4 22 4 4 4
31. 2_{−} 3_{+} 2_{−}  -1  4 2 1^{2} 3_{+} 2  2 1^{2} 1_{i} 31∗ 31∗ 4 2 22 4 4 22 2 22
D_{4}  3  4  1^{2} 1_{i} 31∗ 31∗ 4 4 22 4 22 4 2 4
32. 2_{−} 5_{−} 2_{+}  0  1 0 1^{2} 3_{i} −2 −2 2^{2} 2_{+} 1  V 4 2  4 4 22 2 22
D_{4}  3  8  1^{2} 2_{i} −1 −1  4 4 22 4  4 2 4
33. 0_{+} 9_{−} 2_{−}  0 -14 0 1^{4} 9_{−} 2 −1 1^{2} 1_{k} 31 31 2 2 22 2 4 22 4 22
D_{4}  18  1^{2} 1_{k} 31 31 4 4 22 2 22 4 4 4

Discriminant class: 2 (continued)

∞  2 31 a_{1} a_{2} a_{3}  Over p = 2  Over p = 31  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 37 41 43 47 53 59 61
34. 2_{−} 9_{−} 2_{+}  0 14 0 1^{4} 9_{−} 2 −1 2^{2} 2_{+} 1  V 2 2 22 2 4 22 4
D_{4}  4  18  4 4 22 2 22 4 4 4
35. 2_{−} 9_{−} 2_{+}  0 62 0 1^{4} 9_{−} 2 −1 2^{2} 2_{+} 1  V 4 2 22 4 2  4 22
D_{4}  8  1922  2 2  4 22 2 4 2
36. 2_{−} 11_{−} 2_{+}  0 -12 0 1^{4} 11_{−} 2  2 2^{2} 2_{+} 1  V 2 4 22 2 2 22 4 22
D_{4}  8  98  2 2  2  2 4 2
37. 2_{−} 11_{+} 2_{−}  0 12 0 1^{4} 11_{+} 2  2 1^{2} 1_{i} 31∗ 31∗ 2 4  2 2 22 4 22
D_{4} 16  98  1^{2} 1_{i} 31∗ 31∗ 2 2  2 22 2 4 2
38. 2_{−} 11_{−} 2_{+}  0 16 0 1^{4} 11_{−} 2 −2 2^{2} 2_{+} 1  V 4 4 22 4 2 22 2 22
D_{4}  4  2  2 2  4 22 2 2 2
39. 0_{+} 11_{−} 2_{−}  0 -16 0 1^{4} 11_{−} 2 −2 1^{2} 1_{k} 31 31 4 4 22 4 2 22 2
D_{4}  2  1^{2} 1_{k} 31 31 2 2  4 22 2 2 2
40. 2_{−} 11_{−} 2_{+}  0  0 0 1^{4} 11_{−} 2 −2 2^{2} 2_{+} 1  V 2 4 22 2 4  2 22
D_{4}  8  1922  4 4 22 2 22 4 2 4

Discriminant class: 31

∞  2 31 a_{1} a_{2} a_{3}  Over p = 2  Over p = 31  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 37 41 43 47 53 59 61
41. 2_{−} 10_{i} 1_{i}  0 14 0 1^{4} 10_{i} −1 2 1^{2} 1_{i} 31∗ 31∗ 22 22 2 22 4 2 4
D_{4} 4 31 2 0_{+} ∗ ∗ 4 4 22 22 2 4 4 4
42. 0_{+} 10_{i} 1_{k}  0 -14 0 1^{4} 10_{i} −1 2 1^{2} 1_{k} 31 31 22 22 2 22 4 2 4 22
D_{4} 31 4 4 22 22 2 4 4 4
43. 2_{−} 2_{i} 3_{i}  0 0 0 1^{2} 2_{i} −1 −1 1^{4} 3_{i} 31 31∗ 22 22 2 22 4 4 2 22
D_{4}  3  31  4 4  22 2 4 2 4
44. 2_{−} 6_{i} 3_{i}  0 0 0 1^{2} 3_{i} −2 −2 1^{4} 3_{i} 31 31∗ 22  2 22 4 4 2 22
D_{4} 12  124  1^{2} 3_{+} 2  2  4 4  22 2 4 2 4
45. 2_{−} 10_{i} 3_{i}  0 62 0 1^{4} 10_{i} −1 2 1^{4} 3_{i} 31 31∗ 22 22 2 22 2 4 4 22
D_{4} 16  1519  2 2 22  2 2 4 2
46. 2_{−} 10_{i} 3_{i}  0 -62 0 1^{4} 10_{i} −1 2 1^{4} 3_{i} 31 31∗  22 2  2 4 4 22
D_{4} 16  1519  2 2 22 22 2 2 4 2

Discriminant class: 62

∞  2 31 a_{1} a_{2} a_{3}  Over p = 2  Over p = 31  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 37 41 43 47 53 59 61
47. 2_{−} 11_{i} 1_{i}  0 16 0 1^{4} 11_{i} −2 2 1^{2} 1_{i} 31∗ 31∗ 4 4 2 4 22 2 22 22
D_{4} 4 62 2 0_{+} ∗ ∗ 22 22 4 2 22 22 22
48. 0_{+} 11_{i} 1_{k}  0 -16 0 1^{4} 11_{i} −2 2 1^{2} 1_{k} 31 31 4 4 2 4 22 2 22
D_{4} 62 22 22 4 2 22 22 22
49. 2_{−} 3_{i} 3_{i} -1 12 -2 1^{2} 3_{i} −2 −2 1^{4} 3_{i} 31 31∗ 4 2 2 4 22 4  22
D_{4}  3  4  22 22 22 4 2 22 22 22
50. 2_{−} 5_{i} 3_{i}  0 -31 0 1^{2} 3_{+} 2  2 1^{4} 3_{i} 31 31∗ 4 2 2 4 22 4 22 22
D_{4} 12  248  1^{2} 2_{i} −1 −1  22 22 22 4 2 22  22
51. 2_{−} 11_{i} 3_{i}  0 0 0 1^{4} 11_{i} −2 2 1^{4} 3_{i} 31 31∗ 2 4 2 2 22 4 22 22
D_{4} 16  248  22 22 22 2 2 22 22

Discriminant class: 62 (continued)

∞  2 31 a_{1} a_{2} a_{3}  Over p = 2  Over p = 31  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 37 41 43 47 53 59 61
52. 2_{−} 11_{i} 3_{i}  0 0 0 1^{4} 11_{i} −2 2 1^{4} 3_{i} 31 31∗ 2 4 2 2  4 22 22
D_{4} 80  62  22 2 2  22 22