Quartic Fields Unramified Outside of {∞, 2, 37}

Discriminant class: −74

∞  2 37 a_{1} a_{2} a_{3}  Over p = 2  Over p = 37  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 41 43 47 53 59 61
1. 1_{k} 11_{i} 3_{+}  0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{+} 37∗ 37∗ 22  2 22 22 4 2 22
D_{4} 10  -296  22 22 22 2 2 4 2 22
2. 1_{k} 11_{i} 3_{+}  0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{+} 37∗ 37∗ 22 22 2 22  4 2 22
D_{4}  2  -74  22 22 2 2 4 2

Discriminant class: −37

∞ 2 37 a_{1} a_{2} a_{3}  Over p = 2  Over p = 37 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 41 43 47 53 59 61
3. 1_{k} 4_{k} 3_{−}  0 0 0 2^{2} 4_{k} −∗ ∗ 1^{4} 3_{−} 37 37 2 4 2 2 4 4 22 22
D_{4} -37 4 22 22 2 22 22 4
4. 1_{k} 6_{k} 3_{−}  0 0 0 2^{2} 6_{k} −∗ ∗ 1^{4} 3_{−} 37 37 2 4 2 2 4 4 22 22
D_{4} -148 4 22 22 2 22 4

Discriminant class: −2

∞  2 37  a_{1} a_{2} a_{3}  Over p = 2  Over p = 37 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 41 43 47 53 59 61
5. 1_{k} 11_{i} 0_{+}  0 0 0 1^{4} 11_{i} −2 2 4 0_{+} ∗ ∗ 22 4 2 22 4 22 22 2
D_{4}  -2  4 2 22 22 2 4 22 4
6. 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  -2738  4 2 22 22 2 4 22 4

Discriminant class: −1

∞  2 37  a_{1} a_{2} a_{3}  Over p = 2  Over p = 37  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 41 43 47 53 59 61
7. 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  2 0_{+} ∗  ∗ 22 2  4 2 22 4 22
8. 1_{k} 4_{k} 2_{−}  0 1 0 2^{2} 4_{k} −1  ∗ 2^{2} 2_{−} 1  V 2 22 2 2 22 22 4 4
D_{4}  -9  22 4 22 4 2 22 4 22
9. 1_{k} 6_{i} 2_{+}  0 2 0 2^{2} 6_{i} −1  ∗ 1^{2} 1_{+} 37 37 2 22 2 2 22 22 4 4
D_{4}  -36  1^{2} 1_{+} 37 37 22 4 22 4 2  4 22
10. 1_{k} 10_{k} 2_{−}  0 74 0 1^{4} 10_{k} −1  2 2^{2} 2_{−} 1  V 4 22 2 4 22  4 2
D_{4}  2  -1369  22 2  4 2 22 4 22
11. 1_{k} 10_{k} 2_{−}  0 10 0 1^{4} 10_{k} −1 2∗ 2^{2} 2_{−} 1  V 4 22 2 4 22 22 2 4
D_{4}  2  -49  4 22 2 2 22 2
12. 1_{k} 10_{i} 2_{+}  0 -14 0 1^{4} 10_{i} −1 2∗ 1^{2} 1_{−} 37∗ 37∗ 4  2 4  22 2 4
D_{4}  2  -25  1^{2} 1_{−} 37∗ 37∗ 22 4 22 2 2 22 2 22
13. 1_{k} 2_{i} 2_{+}  -1 -1 4 1^{2} 2_{i} −∗ −∗ 1^{3} 2_{+} 1  4 3 2 4 22 3 4 4
S_{4}  2  2 0_{+} ∗  ∗  3 4 3 2 4 22 2 3

Discriminant class: −1 (continued)

∞  2 37  a_{1} a_{2} a_{3}  Over p = 2  Over p = 37  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 41 43 47 53 59 61
14. 1_{k} 6_{i} 2_{+}  2 -9 -36 1^{2} 3_{i} −2 −2 1^{3} 2_{+} 1  4 3 4 2 22 3 2 4
S_{4}  -34  1^{2} 3_{+} 2  2  3 2 3 4 2  2 3
15. 1_{k} 6_{i} 2_{+}  2 3 -4 1^{2} 3_{−} 2∗  2∗ 1^{3} 2_{+} 1  2 3 4 4 22 3 4 2
S_{4}  -12  1^{2} 3_{k} −2∗ −2∗  3 4 3 4 4 22 2 3

Discriminant class: 1

∞ 2 37 a_{1} a_{2} a_{3}  Over p = 2  Over p = 37  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 41 43 47 53 59 61
16. 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
17. 2_{−} 4_{−} 2_{+}  0 19 0 2^{2} 4_{−} 1 {−1 \atop −∗} 1^{2} 1_{+} 37 37 22 22 22 22 22 22 22 22
V_{4}  81  1^{2} 1_{+} 37 37 22 22  22 22  22 22
18. 2_{−} 6_{+} 2_{−}  2 -13 -14 2^{2} 6_{+} 1 {−2 \atop −2∗} 2^{2} 2_{−} 1  V  22 22  22 22 22 22
V_{4} 5  123  22 22  22 22 22 22 22
19. 0_{+} 6_{−} 2_{−}  -2 -21 22 2^{2} 6_{−} 1  {∗ \atop 2} 2^{2} 2_{−} 1  V 22 22  22 22 22 22 22
V_{4}  47  22 22  22  22 22 22
20. 2_{−} 8_{+} 2_{−}  0 36 0 1^{4} 8_{+} 1 {−2∗ \atop −∗} 2^{2} 2_{−} 1  V 22 22 22 22 22 22 22
V_{4} 10  361  22  22 22 22 22 22
21. 2_{−} 8_{−} 2_{+}  0 0 0 1^{4} 8_{−} 1 {−2∗ \atop −1} 1^{2} 1_{−} 37∗ 37∗ 22  22 22  22 22 22
V_{4} 10  1369  1^{2} 1_{−} 37∗ 37∗  22  22 22 22 22
22. 2_{−} 8_{+} 2_{−}  0 -36 0 1^{4} 8_{+} 1 {−2 \atop −∗} 2^{2} 2_{−} 1  V 22 22 22 22 22 22  22
V_{4} 2  361  22 22  22 22  22
23. 2_{−} 6_{−} 2_{+}  0 -8 -8 1^{4} 6_{−} 1  1^{3} 2_{+} 1  3 3 3  3 3 3 22
A_{4}  44  22 3 22 22 3 3 3

Discriminant class: 2

∞  2 37 a_{1} a_{2} a_{3}  Over p = 2  Over p = 37  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 41 43 47 53 59 61
24. 2_{−} 11_{−} 0_{+}  0  4 0 1^{4} 11_{−} 2 2 4 0_{+} ∗  ∗ 4 4  4 4  4
C_{4}  2  4 22 22 4 22 4 4 4
25. 0_{+} 11_{+} 0_{+}  0 -4 0 1^{4} 11_{+} 2 2 4 0_{+} ∗  ∗ 4 4 22 4 4  4 22
C_{4}  2  4  22 4  4 4 4
26. 0_{+} 11_{+} 2_{+}  0 -148 0 1^{4} 11_{+} 2 2 2^{2} 2_{+} ∗  ∗ 4 4 22 4 4 22 4
C_{4}  2  2738  4 22 22 4  4 4 4
27. 2_{−} 11_{−} 2_{+}  0 148 0 1^{4} 11_{−} 2 2 2^{2} 2_{+} ∗  ∗ 4 4  4 4 22 4 22
C_{4} 26  2738  4  22 4 22 4 4 4
28. 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 22  4 22 2 4 2
29. 2_{−} 11_{−} 0_{+}  0  0 0 1^{4} 11_{−} 2 −2 4 0_{+} ∗  ∗ 2 4 22 2 4 22 2 22
D_{4}  2  4 22 22 2 22 4 2 4
30. 2_{−} 9_{+} 2_{−}  0 74 0 1^{4} 9_{+} 2 −1 1^{2} 1_{−} 37∗ 37∗ 4 2 22 4 2  4 22
D_{4}  2  2738  1^{2} 1_{+} 37 37 2 22  4 22 2 4 2

Discriminant class: 2 (continued)

∞  2 37 a_{1} a_{2} a_{3}  Over p = 2  Over p = 37  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 41 43 47 53 59 61
31. 2_{−} 11_{−} 2_{+}  0  0 0 1^{4} 11_{−} 2 −2 2^{2} 2_{+} ∗  ∗ 2 4 22 2 4  2 22
D_{4}  2  2738  4 22 22 2 22 4 2 4

Discriminant class: 37

∞ 2 37 a_{1} a_{2} a_{3}  Over p = 2  Over p = 37  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 41 43 47 53 59 61
32. 2_{−} 0_{+} 3_{−}  -1 5 -7 4 0_{+} ∗ ∗ 1^{4} 3_{−} 37 37 22 4  22 4 4 4 4
C_{4}  49  4 4 22 4  4 4
33. 0_{+} 4_{−} 3_{−}  0 -37  0 2^{2} 4_{−} ∗ ∗ 1^{4} 3_{−} 37 37  4 22  4 4 4 4
C_{4}  333  4 4 22 4 22  4 4
34. 2_{−} 6_{+} 3_{−}  0 74  0 2^{2} 6_{+} ∗ ∗ 1^{4} 3_{−} 37 37  4  4 4 4 4
C_{4} 50  1332  4 4 22 4  22 4 4
35. 0_{+} 6_{−} 3_{−}  0 -74  0 2^{2} 6_{−} ∗ ∗ 1^{4} 3_{−} 37 37 22 4 22 22 4 4 4 4
C_{4}  1332  4 4 22 4 22 22 4 4
36. 2_{−} 6_{+} 1_{−}  -2 -4  6 1^{4} 6_{+} ∗ −1 1^{2} 1_{−} 37∗ 37∗ 22 2 22 22 2 2 4 4
D_{4}  9  2 0_{+} ∗  ∗ 2 4 22 4 22 22 4 2
37. 2_{−} 8_{−} 1_{+}  0 -12  0 1^{4} 8_{−} ∗ −1 1^{2} 1_{+} 37 37 22 2 22 22 2 2 4 4
D_{4}  37  2 4 22 4 22  4 2
38. 2_{−} 6_{+} 3_{−}  0 0  0 1^{4} 6_{+} ∗ −∗ 1^{4} 3_{−} 37 37 22 4 22 22 4 4 2 2
D_{4} 4  148  4 2  2 22 22 2 4
39. 2_{−} 8_{+} 3_{−}  0 0  0 1^{4} 8_{+} ∗ −∗ 1^{4} 3_{−} 37 37 22 4 22 22 4 4 2 2
D_{4} 4  37  4 2  2 22  2 4
40. 2_{−} 4_{+} 1_{−}  0 2  2 1^{4} 4_{+} ∗  1^{2} 1_{−} 37∗ 37∗ 3 4 3 3 4 4 4 2
S_{4}  1  2 0_{+} ∗  ∗ 2 2 3 4 3 3 2 2
41. 2_{−} 8_{+} 1_{−}  0 2 -4 1^{4} 8_{+} ∗  1^{2} 1_{−} 37∗ 37∗ 3 2 3 3 4 4 2 2
S_{4}  3  2 0_{+} ∗  ∗ 4 2 3 4 3 3 4 4
42. 2_{−} 8_{−} 1_{+}  0 4 -4 1^{4} 8_{−} ∗  1^{2} 1_{+} 37 37 3 4 3 3 2 2 4 2
S_{4}  2  4 2 3 2 3 3 4 4
43. 2_{−} 2_{+} 3_{−}  -1 5 -7 1^{3} 2_{+} ∗  1^{4} 3_{−} 37 37 3 2 3 3 2 4 4 4
S_{4} 8  12  4 2 3 2 3 3 2 2
44. 2_{−} 4_{+} 3_{−}  2 20 -18 1^{4} 4_{+} ∗  1^{4} 3_{−} 37 37 3 4 3 3 4 2 2 4
S_{4} 4  7  4 2 3 4 3 3 2 2
45. 0_{+} 8_{−} 3_{−}  0 -74 148 1^{4} 8_{−} ∗  1^{4} 3_{−} 37 37 3 4 3 3 2 4 2 4
S_{4} 2  407  2 2 3 2 3 3 4 4
46. 2_{−} 8_{+} 3_{−}  0 0 -148 1^{4} 8_{+} ∗  1^{4} 3_{−} 37 37 3 2 3 3 4 2 4 4
S_{4} 12  370  2 2 3 4 3 3 4 4

Discriminant class: 74

∞  2 37 a_{1} a_{2} a_{3}  Over p = 2  Over p = 37  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 41 43 47 53 59 61
47. 2_{−} 11_{−} 3_{+}  0 148 0 1^{4} 11_{−} 2∗ 2∗ 1^{4} 3_{+} 37∗ 37∗ 4  22 4 22 4 22 4
C_{4} 68  3626  4  4
48. 0_{+} 11_{+} 3_{+}  0 -148 0 1^{4} 11_{+} 2∗ 2∗ 1^{4} 3_{+} 37∗ 37∗ 4  4 22 4  4
C_{4}  2  3626  4  22 22 4 22

Discriminant class: 74 (continued)

∞  2 37 a_{1} a_{2} a_{3}  Over p = 2  Over p = 37  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 41 43 47 53 59 61
49. 0_{+} 11_{+} 3_{+}  0 -148 0 1^{4} 11_{+} 2∗ 2∗ 1^{4} 3_{+} 37∗ 37∗ 4 22  4  4 22 4
C_{4}  2  1850  22 4  22 4  22
50. 2_{−} 11_{−} 3_{+}  0 148 0 1^{4} 11_{−} 2∗ 2∗ 1^{4} 3_{+} 37∗ 37∗ 4 22 22 4  4  4
C_{4} 36  1850  22 4  22  4 22 22
51. 2_{−} 9_{+} 1_{−}  0 10 0 1^{4} 9_{+} 2∗ −1 1^{2} 1_{−} 37∗ 37∗ 4  22 4  2 22 4
D_{4}  74  22 4 22 22 22 2 22 22
52. 2_{−} 9_{−} 1_{+}  0 14 0 1^{4} 9_{−} 2∗ −1 1^{2} 1_{+} 37 37 4 22 22 4 22 2 22 4
D_{4}  74  2 0_{+} ∗  ∗  4 22 22 22 2 22
53. 2_{−} 11_{−} 3_{+}  0  0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{+} 37∗ 37∗ 2  22 2 22 4 22 2
D_{4} 100  74  22 2 22 22 22 4 22 22
54. 2_{−} 11_{−} 3_{+}  0  0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{+} 37∗ 37∗ 2 22 22 2  4 22 2
D_{4} 20  296  2 22 22 22 4 22