Quartic Fields Ramifying Above {2, 29}

Discriminant class: −58

∞  2 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 29  3 5 7 11 13 17 19 23
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} 3_{+}  0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{+} 29∗ 29∗ 2 4 2 2 4 4 2 2
D_{4} 2 -58 22 22 4 2 22 4 22
2. 1_{k} 11_{i} 3_{+}  0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{+} 29∗ 29∗ 2 4 2 2 4 4 2 2
D_{4} 6 -232 22 4 2 22 4 22 22

Discriminant class: −29

∞  2 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 29  3 5 7 11 13 17 19 23
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
3. 1_{k} 4_{k} 3_{−}  0 0  0 2^{2} 4_{k} −∗  ∗ 1^{4} 3_{−} 29 29 22 22 2 22  4 22 2
D_{4}  -29  22 4 4 22 22 22 2 4
4. 1_{k} 6_{k} 3_{−}  0 0  0 2^{2} 6_{k} −∗  ∗ 1^{4} 3_{−} 29 29 22  2 22 22 4 22 2
D_{4}  -116  22 4 4 22 22  2 4
5. 1_{k} 6_{k} 1_{−} -2 1  2 2^{2} 6_{k} −∗  ∗ 1^{2} 1_{−} 29∗ 29∗ 3 3 4 3 3 4 22 4
S_{4}  -3  2 0_{+} ∗  ∗ 3 2 4 3 3 3 4 4
6. 1_{k} 10_{k} 1_{−}  0 -6 -8 1^{4} 10_{k} −∗ −2∗ 1^{2} 1_{−} 29∗ 29∗ 3 3 4 3 3 2 22 2
S_{4} -5 2 0_{+} ∗ ∗ 3 2 2 3 3 3 2 2
7. 1_{k} 10_{i} 1_{+}  0 -2 -8 1^{4} 10_{i} −∗ −2 1^{2} 1_{+} 29 29 3 3 2 3 3 4 22 4
S_{4}  11  3 2 4 3 3 3 4 4
8. 1_{k} 4_{k} 3_{−}  2 16 44 2^{2} 4_{k} −∗  ∗ 1^{4} 3_{−} 29 29 3 3 4 3 3 2  2
S_{4}  2  20  3 2 4 3 3 3 2 2
9. 1_{k} 6_{k} 3_{−} -2 -13 -44 1^{2} 3_{+} 2  2 1^{4} 3_{−} 29 29 3 3 2 3 3 4 22 4
S_{4} 2 252 1^{2} 3_{k} −2∗ −2∗ 3 2 2 3 3 3 4 4
10. 1_{k} 10_{k} 3_{−}  0 -58 -232 1^{4} 10_{k} −∗ −2∗ 1^{4} 3_{−} 29 29 3 3 4 3 3 4 22 4
S_{4} 4 -261 3 2 2 3 3 3 4 4
11. 1_{k} 10_{k} 3_{−}  0 -58 -464 1^{4} 10_{k} −∗ −2 1^{4} 3_{−} 29 29 3 3 2 3 3 2 22 2
S_{4} 4 -725 3 2 4 3 3 3 2 2

Discriminant class: −2

∞  2 29  a_{1} a_{2} a_{3}  Over p = 2  Over p = 29 3 5 7 11 13 17 19 23
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
12. 1_{k} 11_{i} 2_{+}  0 0  0 1^{4} 11_{i} −2  2 2^{2} 2_{+} ∗ ∗ 22 4 2 22 4  22 2
D_{4}  2  -1682  2 4  22 2 4 22 4
13. 1_{k} 9_{i} 2_{+}  0 4 -16 1^{4} 9_{i} −2 −∗ 1^{3} 2_{+} ∗  3 2 4 3 4  3 4
S_{4}  -40  2 2 3 3 2 4 22 4
14. 1_{k} 11_{i} 2_{+}  0 -4 40 1^{4} 11_{i} −2  2 1^{3} 2_{+} ∗  3 2 2 3 4 22 3 4
S_{4}  2  18  2 2 3 3 2 4 22 2
15. 1_{k} 11_{i} 2_{+}  0 -20 -112 1^{4} 11_{i} −2 −2∗ 1^{3} 2_{+} ∗  3 2 4 3 2 22 3 2
S_{4}  -130  2 2 3 3 2 2 22 4

Discriminant class: −1

∞  2 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 29  3 5 7 11 13 17 19 23
G  h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
16. 1_{k} 4_{k} 2_{−}  0 -5 0 2^{2} 4_{k} −1 ∗ 2^{2} 2_{−} 1  V 4  2 4 22 22 4 2
D_{4}  -1  4 22 22 4 4 22 2 22
17. 1_{k} 6_{i} 2_{+}  0 10 0 2^{2} 6_{i} −1 ∗ 1^{2} 1_{+} 29 29 4 22 2 4  22 4 2
D_{4}  -4  1^{2} 1_{+} 29 29 4 22 22 4 4  2 22
18. 1_{k} 10_{i} 2_{+}  0 -14 0 1^{4} 10_{i} −1 2∗ 1^{2} 1_{−} 29∗ 29∗ 2 22 2 2 22 22 2 2
D_{4}  2  -9  1^{2} 1_{−} 29∗ 29∗ 4 22 22 2 4 22 4
19. 1_{k} 10_{k} 2_{−}  0 58 0 1^{4} 10_{k} −1 2 2^{2} 2_{−} 1  V 4 22 2 4 22  4 2
D_{4}  2  -841  2 22 22 4 2 22 4 22
20. 1_{k} 10_{k} 2_{−}  0 6 0 1^{4} 10_{k} −1 2∗ 2^{2} 2_{−} 1  V 2 22 2 2 22 22 2 2
D_{4}  2  -49  4  22 2 4 22 4 22

Discriminant class: 1

∞ 2 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 29  3 5 7 11 13 17 19 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
21. 2_{−} 4_{−} 2_{+} 0 15 0 2^{2} 4_{−} 1 {−1 \atop −∗} 1^{2} 1_{+} 29 29 22  22 22  22 22 22
V_{4} 3  49  1^{2} 1_{+} 29 29 22 22 22 22 22  22 22
22. 0_{+} 6_{−} 2_{−} -2 -17 18 2^{2} 6_{−} 1  {∗ \atop 2} 2^{2} 2_{−} 1  V 22 22  22 22 22 22
V_{4}  23  22 22 22 22 22 22 22 22
23. 2_{−} 6_{+} 2_{−} 2 -9 -10 2^{2} 6_{+} 1 {−2 \atop −2∗} 2^{2} 2_{−} 1  V 22 22 22 22 22 22 22 22
V_{4}  83  22 22 22 22 22 22  22
24. 2_{−} 8_{+} 2_{−} 0 -28 0 1^{4} 8_{+} 1 {−2 \atop −∗} 2^{2} 2_{−} 1  V  22 22  22 22  22
V_{4} 6  225  22 22 22  22 22 22 22
25. 2_{−} 8_{−} 2_{+} 0 0 0 1^{4} 8_{−} 1 {−2∗ \atop −1} 1^{2} 1_{−} 29∗ 29∗ 22 22 22 22 22 22 22 22
V_{4} 2 841 1^{2} 1_{−} 29∗ 29∗ 22 22 22 22 22 22
26. 2_{−} 8_{+} 2_{−} 0 28 0 1^{4} 8_{+} 1 {−2∗ \atop −∗} 2^{2} 2_{−} 1  V 22 22 22 22 22 22 22 22
V_{4} 6  225  22 22 22  22 22 22

Discriminant class: 2

∞  2 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 29  3 5 7 11 13 17 19 23
G  h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
27. 0_{+} 11_{+} 2_{+}  0 -116 0 1^{4} 11_{+} 2 2 2^{2} 2_{+} ∗  ∗ 4 4 22 4 4 22 4 22
C_{4}  2  1682  22 4  4 22 4 4 4
28. 2_{−} 11_{−} 2_{+}  0 116 0 1^{4} 11_{−} 2 2 2^{2} 2_{+} ∗  ∗ 4 4  4 4 22 4
C_{4} 34  1682  4  4  4 4 4
29. 2_{−} 9_{+} 2_{−}  0 58 0 1^{4} 9_{+} 2 −1 1^{2} 1_{−} 29∗ 29∗ 4 2 22 4 2  4 22
D_{4}  2  1682  1^{2} 1_{+} 29 29 22 2 22 4 22 2 4 2
30. 2_{−} 11_{−} 2_{+}  0  0 0 1^{4} 11_{−} 2 −2 2^{2} 2_{+} ∗  ∗ 2 4 22 2 4  2 22
D_{4}  2  1682  22 4  2 22 4 2 4
31. 2_{−} 3_{−} 2_{+}  -1  0 -7 1^{2} 3_{−} 2∗ 2∗ 1^{3} 2_{+} ∗  2 4 3 4 4 3 4 3
S_{4}  9  2 0_{+} ∗ ∗  3 2 22 4 3 4 4 4
32. 0_{+} 9_{+} 2_{+}  0 -20 4 1^{4} 9_{+} 2 −1 1^{3} 2_{+} ∗  4 2 3 4 4 3 2 3
S_{4}  73  3 4 22 4 3 2 2 2

Discriminant class: 2 (continued)

∞  2 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 29  3 5 7 11 13 17 19 23
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_{−} 9_{−} 2_{+}  0 -6 -8 1^{4} 9_{−} 2 −1 1^{3} 2_{+} ∗  4 4 3 2 2 3 4 3
S_{4}  2  26  3 4 22 2 3 4 4 4
34. 2_{−} 11_{−} 2_{+}  0 36 -32 1^{4} 11_{−} 2 2 1^{3} 2_{+} ∗  2 2 3 2 4 3 2 3
S_{4} 16  66  3 2 22 4 3 2 4 4
35. 2_{−} 11_{−} 2_{+}  0  4 -16 1^{4} 11_{−} 2 −2 1^{3} 2_{+} ∗  4 2 3 4 2 3 2 3
S_{4}  2  76  3 4 22 2 3 2 4 4
36. 2_{−} 11_{−} 2_{+}  0 20 -48 1^{4} 11_{−} 2 −2 1^{3} 2_{+} ∗  4 4 3 2 4 3 4 3
S_{4}  2  44  3 4 22 4 3 4 2 2
37. 2_{−} 11_{−} 2_{+}  0  4 -16 1^{4} 11_{−} 2 2 1^{3} 2_{+} ∗  2 4 3 4 2 3 4 3
S_{4}  8  18  3 2  2 3 4 2 2

Discriminant class: 29

∞ 2 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 29  3 5 7 11 13 17 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
38. 0_{+} 4_{−} 3_{−} 0 -29 0 2^{2} 4_{−} ∗ ∗ 1^{4} 3_{−} 29 29 4 22 22 4 22 4 4 22
C_{4}  29  4 4 4 4 4  22 4
39. 0_{+} 6_{−} 3_{−} 0 -58 0 2^{2} 6_{−} ∗ ∗ 1^{4} 3_{−} 29 29 4  22 4  4 4 22
C_{4}  116  4 4 4 4 4 22  4
40. 2_{−} 6_{+} 3_{−} 0 58 0 2^{2} 6_{+} ∗ ∗ 1^{4} 3_{−} 29 29 4  4  4 4
C_{4} 26  116  4 4 4 4 4 22 22 4
41. 2_{−} 6_{+} 1_{−} 2 4 8 1^{4} 6_{+} ∗ −1 1^{2} 1_{−} 29∗ 29∗ 4  22 4 22 2 4 22
D_{4} 10 2 0_{+} ∗ ∗ 4 2 2 4 4 22 22 2
42. 2_{−} 8_{−} 1_{+} 0 4 0 1^{4} 8_{−} ∗ −1 1^{2} 1_{+} 29 29 4 22 22 4  2 4 22
D_{4} 29 4 2 2 4 4 22 2
43. 2_{−} 6_{+} 3_{−} -2 16 14 1^{4} 6_{+} ∗ −∗ 1^{4} 3_{−} 29 29 2 22 22 2  4 2 22
D_{4} 12 49 2 4 4 2 2 22 22 4
44. 2_{−} 8_{+} 3_{−} 0 0 0 1^{4} 8_{+} ∗ −∗ 1^{4} 3_{−} 29 29 2  22 2 22 4 2 22
D_{4} 12  29  2 4 4 2 2  22 4

Discriminant class: 58

∞  2 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 29  3 5 7 11 13 17 19 23
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
45. 0_{+} 11_{+} 3_{+}  0 -116 0 1^{4} 11_{+} 2∗ 2∗ 1^{4} 3_{+} 29∗ 29∗ 22 4  4 4 22
C_{4}  2  522  4  4  4 4 4
46. 0_{+} 11_{+} 3_{+}  0 -116 0 1^{4} 11_{+} 2∗ 2∗ 1^{4} 3_{+} 29∗ 29∗  4  22 4 4
C_{4}  2  2842  4 22 4 22 4 4 4 22
47. 2_{−} 11_{−} 3_{+}  0 116 0 1^{4} 11_{−} 2∗ 2∗ 1^{4} 3_{+} 29∗ 29∗  4 22 22 4 4  22
C_{4} 68  522  4  4 22 4 4 4
48. 2_{−} 11_{−} 3_{+}  0 116 0 1^{4} 11_{−} 2∗ 2∗ 1^{4} 3_{+} 29∗ 29∗ 22 4 22  4 4 22 22
C_{4} 20  2842  4 22 4  4 4 4 22
49. 2_{−} 9_{−} 1_{+}  0 14 0 1^{4} 9_{−} 2∗ −1 1^{2} 1_{+} 29 29 22 2 22 22 2 2 22 22
D_{4} 58 2 0_{+} ∗ ∗ 4 2 22 4 2 4 22
50. 2_{−} 9_{+} 1_{−}  0 -6 0 1^{4} 9_{+} 2∗ −1 1^{2} 1_{−} 29∗ 29∗ 22 2 22 22 2 2 22 22
D_{4}  58  4 22 2 22 4 2 4

Discriminant class: 58 (continued)

∞  2 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 29  3 5 7 11 13 17 19 23
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_{−} 11_{−} 3_{+}  0  0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{+} 29∗ 29∗ 22 4 22 22 4 4 22 22
D_{4}  4  232  2 22 4 22 2 4 2
52. 2_{−} 11_{−} 3_{+}  0  0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{+} 29∗ 29∗ 22 4 22 22 4 4 22 22
D_{4} 12  58  2  4 22 2 4 2 22