Quartic Fields Unramified Outside of {∞, 2, 43}

Discriminant class: −86

∞  2 43 a_{1} a_{2} a_{3}  Over p = 2  Over p = 43  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 47 53 59 61
1. 1_{k} 11_{−} 3_{k}  0 0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{k} 43 43∗ 22 22 2 2 4  22 22
D_{4}  -344  22 22  22 4 2 22
2. 1_{k} 11_{−} 3_{k}  0 0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{k} 43 43∗ 22  2 2 4  22 22
D_{4}  -86  22 22  22 4 2

Discriminant class: −43

∞ 2 43 a_{1} a_{2} a_{3}  Over p = 2  Over p = 43  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 47 53 59 61
3. 1_{k} 6_{+} 3_{i}  0 0 0 1^{4} 6_{+} ∗ −∗ 1^{4} 3_{i} 43∗ 43 2 4 2 22  22 2 22
D_{4}  -172  4 22 4 22 22  22 4
4. 1_{k} 8_{+} 3_{i}  0 0 0 1^{4} 8_{+} ∗ −∗ 1^{4} 3_{i} 43∗ 43 2 4 2 22 22 22 2 22
D_{4} 3  -43  4 22 4 22 22 22 22 4
5. 1_{k} 4_{+} 1_{i}  0 0 -2 1^{4} 4_{+} ∗  1^{2} 1_{i} 43∗ 43∗ 4 4 4 3 3 3 2 3
S_{4}  -1  4 3 2 3 22 3 22 4
6. 1_{k} 2_{+} 3_{i}  -1 -5 -35 1^{3} 2_{+} ∗  1^{4} 3_{i} 43∗ 43 2 4 4 3 3 3 4 3
S_{4}  64  2 3 2 3 22 3 22 2
7. 1_{k} 4_{+} 3_{i}  -2 -20 64 1^{4} 4_{+} ∗  1^{4} 3_{i} 43∗ 43 4 2 2 3 3 3 4 3
S_{4} 2  -8  4 3 2 3  3 22 4

Discriminant class: −2

∞  2 43  a_{1} a_{2} a_{3}  Over p = 2  Over p = 43  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 47 53 59 61
8. 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  2 0_{+} ∗  ∗ 4 2 4 22 2 4 22 4
9. 1_{k} 9_{k} 2_{−}  0 10 0 1^{4} 9_{k} −2 −∗ 1^{2} 1_{k} 43 43 22 4 2 22 2  4
D_{4}  -18  1^{2} 1_{k} 43 43 4 4 4 22 4 2 22 4
10. 1_{k} 9_{i} 2_{+}  0 -10 0 1^{4} 9_{i} −2 −∗ 2^{2} 2_{+} 1  V  4 2 22 2  22 4
D_{4}  -18  4 4 4 22 4 2 22 4
11. 1_{k} 11_{i} 2_{+}  0 0 0 1^{4} 11_{i} −2  2 2^{2} 2_{+} 1  V 22 4 2 22 4 22 22 2
D_{4}  -3698  4 2 4 22 2 4 22 4
12. 1_{k} 11_{i} 2_{+}  0 12 0 1^{4} 11_{i} −2 −2∗ 2^{2} 2_{+} 1  V 22 2 2 22 4 22 22 4
D_{4}  -50  2 4 2  4 4  2
13. 1_{k} 11_{k} 2_{−}  0 -12 0 1^{4} 11_{k} −2 −2∗ 1^{2} 1_{i} 43∗ 43∗ 22 2 2  4 22 22 4
D_{4}  -50  1^{2} 1_{i} 43∗ 43∗ 2 4 2  4 4 22 2
14. 1_{k} 9_{i} 2_{+}  0 -16 -4 1^{4} 9_{i} −2 −∗ 1^{3} 2_{+} 1  22 4 2 3 2 3 3 2
S_{4}  -7  2 4 2 3 2 2 3 2
15. 1_{k} 11_{i} 2_{+}  0 8 -16 1^{4} 11_{i} −2  2 1^{3} 2_{+} 1  22 4 4 3 2 3 3 2
S_{4}  2  -34  2 4 4 3 4 2 3 2
16. 1_{k} 11_{i} 2_{+}  0 -40 32 1^{4} 11_{i} −2 −2∗ 1^{3} 2_{+} 1  22 2 4 3 2 3 3 2
S_{4}  440  2 2 4 3 4 2 3 2

Discriminant class: −1

∞  2 43  a_{1} a_{2} a_{3}  Over p = 2  Over p = 43 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 47 53 59 61
17. 1_{k} 10_{i} 0_{+}  0 -2 0 1^{4} 10_{i} −1 2 4 0_{+} ∗ ∗ 4 22 2 4 22 22 4 2
D_{4} -1 22 2 22 2 22 4 22
18. 1_{k} 10_{k} 2_{−}  0 86 0 1^{4} 10_{k} −1 2 2^{2} 2_{−} ∗ ∗ 4 22 2 4 22 22 4 2
D_{4}  -1849  22 2 22  2 22 4 22

Discriminant class: 1

∞ 2 43 a_{1} a_{2} a_{3}  Over p = 2  Over p = 43  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 47 53 59 61
19. 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
20. 2_{−} 4_{−} 2_{+}  0 -21 0 2^{2} 4_{−} 1  {−1 \atop −∗} 2^{2} 2_{+} 1  V 22 22 22 22  22 22
V_{4}  121  22 22 22  22  22 22
21. 2_{−} 6_{−} 2_{+}  2 19 18 2^{2} 6_{−} 1  {∗ \atop 2} 2^{2} 2_{+} 1  V 22 22 22 22 22  22
V_{4} 5  167  22  22  22 22 22
22. 2_{−} 6_{+} 2_{−}  2 27 26 2^{2} 6_{+} 1 {−2 \atop −2∗} 1^{2} 1_{i} 43∗ 43∗ 22 22 22  22  22 22
V_{4}  83  1^{2} 1_{i} 43∗ 43∗ 22 22 22  22 22  22
23. 2_{−} 8_{+} 2_{−}  0 44 0 1^{4} 8_{+} 1  {−2 \atop −∗} 1^{2} 1_{k} 43 43  22 22 22 22  22
V_{4} 10  441  1^{2} 1_{k} 43 43 22 22 22  22 22 22 22
24. 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} 10  1849  22  22 22 22
25. 0_{+} 8_{+} 2_{+}  0 -44 0 1^{4} 8_{+} 1 {−2∗ \atop −∗} 2^{2} 2_{+} 1  V 22 22  22 22  22 22
V_{4}  441  22 22 22  22 22 22 22
26. 2_{−} 4_{−} 2_{+}  2 2 -10 2^{2} 4_{−} 1  {−1 \atop −∗} 1^{3} 2_{+} 1  3 3 3  3 3 3 3
A_{4} 2  9  3 3 3 22 22 3 22 3
27. 2_{−} 6_{−} 2_{+}  -2 9 -10 2^{2} 6_{−} 1  {∗ \atop 2} 1^{3} 2_{+} 1  3 3 3 22 3 3 3 3
A_{4} 2  9  3 3 3 22 22 3 22 3
28. 2_{−} 6_{−} 2_{+}  0 -6 -8 1^{2} 3_{k} −2∗ −2∗ 1^{3} 2_{+} 1  3 3 3 22 3 3 3 3
A_{4} 4  40  1^{2} 3_{k} −2∗ −2∗  3 3 3  22 3 22 3
29. 0_{+} 8_{+} 2_{+}  0 -10 -4 1^{4} 8_{+} 1  {−2 \atop −∗} 1^{3} 2_{+} 1  3 3 3 22 3 3 3 3
A_{4} 2  6  3 3 3 22 22 3  3
30. 2_{−} 8_{−} 2_{+}  0 8 -16 1^{4} 8_{−} 1  {−2 \atop −1} 1^{3} 2_{+} 1  3 3 3 22 3 3 3 3
A_{4} 4  9  3 3 3 22  3 22 3

Discriminant class: 2

∞  2 43 a_{1} a_{2} a_{3}  Over p = 2  Over p = 43  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 47 53 59 61
31. 2_{−} 11_{−} 0_{+}  0  4 0 1^{4} 11_{−} 2 2 4 0_{+} ∗  ∗ 4 4  4 4  4
C_{4}  2  4 22 4 22 22 4 4 4
32. 0_{+} 11_{+} 0_{+}  0 -4 0 1^{4} 11_{+} 2 2 4 0_{+} ∗  ∗ 4 4 22 4 4  4 22
C_{4}  2  4  4 22  4 4 4
33. 0_{+} 11_{−} 2_{−}  0 -172 0 1^{4} 11_{−} 2 2 2^{2} 2_{−} ∗  ∗ 4 4 22 4 4  4
C_{4}  3698  4 22 4 22 22 4 4 4

Discriminant class: 2 (continued)

∞  2 43 a_{1} a_{2} a_{3}  Over p = 2  Over p = 43  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 47 53 59 61
34. 2_{−} 11_{+} 2_{−}  0 172 0 1^{4} 11_{+} 2 2 2^{2} 2_{−} ∗  ∗ 4 4  4 4  4 22
C_{4} 50  3698  4  4 22  4 4 4
35. 2_{−} 9_{−} 0_{+}  0 -2 0 1^{4} 9_{−} 2 −1 4 0_{+} ∗  ∗ 4 2 22 4 2 22 4 22
D_{4}  2  2 22 2  22 2 4 2
36. 2_{−} 11_{−} 0_{+}  0  0 0 1^{4} 11_{−} 2 −2 2 0_{+} ∗  ∗ 2 4 22 2 4 22 2 22
D_{4}  2  4 22 4 22 22 4 2 4
37. 2_{−} 9_{+} 2_{−}  0 -86 0 1^{4} 9_{+} 2 −1 2^{2} 2_{−} ∗  ∗ 4 2 22 4 2 22 4 22
D_{4}  2  3698  2 22 2  22 2 4 2
38. 2_{−} 11_{−} 2_{+}  0  0 0 1^{4} 11_{−} 2 −2 1^{2} 1_{i} 43∗ 43∗ 2 4 22 2 4 22 2 22
D_{4}  2  3698  1^{2} 1_{k} 43 43 4 22 4 22 22 4 2 4

Discriminant class: 43

∞  2 43 a_{1} a_{2} a_{3}  Over p = 2  Over p = 43  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 47 53 59 61
39. 2_{−} 10_{k} 1_{k}  0 -10 0 1^{4} 10_{k} −∗ −2 1^{2} 1_{k} 43 43 22 4 22 2 22  4
D_{4}  43  4 4 4 22 4 22 2 4
40. 2_{−} 10_{i} 1_{i}  0 10 0 1^{4} 10_{i} −∗ −2 1^{2} 1_{i} 43∗ 43∗  4 22 2 22  22 4
D_{4}  43  2 0_{+} ∗  ∗ 4 4 4 22 4 22 2 4
41. 2_{−} 4_{k} 3_{k}  0 0 0 2^{2} 4_{k} −∗  ∗ 1^{4} 3_{k} 43 43∗ 22 4 22 2  22 22 2
D_{4}  43  4 2 4 22 2  2 4
42. 2_{−} 6_{k} 3_{k}  0 0 0 2^{2} 6_{k} −∗  ∗ 1^{4} 3_{k} 43 43∗ 22 4 22 2 22 22 22 2
D_{4}  3  172  4 2 4 22 2 22 2 4
43. 0_{+} 10_{i} 3_{k}  0 -86 0 1^{4} 10_{i} −∗ −2∗ 1^{4} 3_{k} 43 43∗ 22 2 22 2 22 22 22 4
D_{4}  1075  2 4 2  4 22 2 2
44. 2_{−} 10_{k} 3_{k}  0 86 0 1^{4} 10_{k} −∗ −2∗ 1^{4} 3_{k} 43 43∗ 22 2  2 22 22 22 4
D_{4} 38  1075  2 4 2  4 22 2 2

Discriminant class: 86

∞  2 43 a_{1} a_{2} a_{3}  Over p = 2  Over p = 43  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 47 53 59 61
45. 2_{−} 11_{k} 1_{k}  0 -12 0 1^{4} 11_{k} −2∗ −2 1^{2} 1_{k} 43 43 2 22 22 22 4 22 2 4
D_{4}  86  2 0_{+} ∗  ∗ 22 4 22  4 4  22
46. 2_{−} 11_{i} 1_{i}  0 12 0 1^{4} 11_{i} −2∗ −2 1^{2} 1_{i} 43∗ 43∗ 2 22 22  4 22 2 4
D_{4}  86  22 4 22  4 4 22 22
47. 2_{−} 9_{i} 3_{i}  0 86 0 1^{4} 9_{i} −2∗ −∗ 1^{4} 3_{i} 43∗ 43 2 22  22 2 22 2 4
D_{4} 38  774  22 4 22  4 2 22 22
48. 0_{+} 9_{k} 3_{i}  0 -86 0 1^{4} 9_{k} −2∗ −∗ 1^{4} 3_{i} 43∗ 43 2 22 22 22 2 22 2 4
D_{4}  774  22 4 22  4 2 22 22
49. 2_{−} 11_{i} 3_{i}  0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{i} 43∗ 43 2  22 22 4  2 2
D_{4} 10  344  22 2  2 4 22
50. 2_{−} 11_{i} 3_{i}  0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{i} 43∗ 43 2 22 22 22 4  2 2
D_{4} 10  86  2 22  2 4 22 22