Quartic Fields Unramified Outside of {∞, 2, 7}

Discriminant class: −14

∞  2 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  3 5 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_{+} 1_{i}  0 -4 0 1^{4} 11_{+} 2 2 1^{2} 1_{i} 7∗ 7∗ 22 22 4 22 2 22 22 4
D_{4} 2 -14 2 4 2 4 2 4 22 22
2. 1_{k} 11_{−} 1_{k} 0 4 0 1^{4} 11_{−} 2 2 1^{2} 1_{k} 7 7 22 22 4 22 2 22  4
D_{4} -14 2 0_{+} ∗ ∗ 2 4 2 4 2 4 22 22
3. 1_{k} 9_{−} 3_{k} 0 14 0 1^{4} 9_{−} 2 −1 1^{4} 3_{k} 7∗ 7 22 22 4 22 4  22 2
D_{4} 2 -14 2 2 4 4 2 2 22 22
4. 1_{k} 9_{−} 3_{k} 0 -14 0 1^{4} 9_{−} 2 −1 1^{4} 3_{k} 7∗ 7  22 4 22 4 22 22 2
D_{4} 2 -14 2 2 4 4 2 2 22
5. 1_{k} 11_{−} 3_{k} 0 0 0 1^{4} 11_{−} 2 −2 1^{4} 3_{k} 7∗ 7 22  2 22 4 22 22 4
D_{4} 4 -56 2 4 4 2 2 4 22
6. 1_{k} 11_{−} 3_{k} 0 0 0 1^{4} 11_{−} 2 −2 1^{4} 3_{k} 7∗ 7 22 22 2  4 22 22 4
D_{4} 2 -14 2 4 4 2 2 4 22 22

Discriminant class: −7

∞ 2 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  3 5 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
7. 1_{k} 6_{−} 1_{k} 2 1 -2 2^{2} 6_{−} 1  {∗ \atop 2} 1^{2} 1_{k} 7 7 4 4 22 4 2 4  22
D_{4}  -1  2 0_{+} ∗ ∗ 2 22 2 22 2 22 4 4
8. 1_{k} 8_{+} 1_{i}  0 -2 4 1^{4} 8_{+} 1 {−2∗ \atop −∗} 1^{2} 1_{i} 7∗ 7∗ 4 4 22 4 2 4 22 22
D_{4} -2 2 22 2 22 2 22 4 4
9. 1_{k} 4_{−} 3_{k} 2 5 4 2^{2} 4_{−} 1 {−1 \atop −∗} 1^{4} 3_{k} 7∗ 7 2 4 22 4 4 2 22 22
D_{4} -3 2 22 4 22 2 2 4
10. 1_{k} 6_{−} 3_{k} 2 5 4 1^{2} 3_{i} −2 −2 1^{4} 3_{k} 7∗ 7 4 2 22 2 4 4 22 22
D_{4} -10 1^{2} 3_{i} −2 −2 2 22 4 22 2 22 4 2
11. 1_{k} 8_{−} 3_{k} 0 -14 -28 1^{4} 8_{−} 1 {−2 \atop −1} 1^{4} 3_{k} 7∗ 7 4 2  2 4 4 22 22
D_{4} 2 -14 2 22 4 2 22 4 2
12. 1_{k} 8_{−} 3_{k} 0 0 0 1^{4} 8_{−} 1 {−2∗ \atop −1} 1^{4} 3_{k} 7∗ 7 2 4 22 4 4 2 22
D_{4} 2 -7 2 4 22 2 22 2 4

Discriminant class: −2

∞  2 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  3 5 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
13. 1_{k} 11_{i} 0_{+} 0 0 0 1^{4} 11_{i} −2 2 2 0_{+} ∗ ∗ 22 4 22 4 22 22 2 4
D_{4} -2 2 4 22 22 2 4 22 4
14. 1_{k} 11_{i} 2_{+} 0 0 0 1^{4} 11_{i} −2 2 1^{2} 1_{k} 7 7 22 4 22 4  22 2 4
D_{4}  2  -98  1^{2} 1_{i} 7∗ 7∗ 2 4  22 2 4 22 4

Discriminant class: −1

∞  2 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  3 5 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
15. 1_{k} 10_{i} 0_{+} 0 -2 0 1^{4} 10_{i} −1 2 2 0_{+} ∗ ∗ 4 22 4 22 22 4 2 22
D_{4} -1 2 22 4 2 22 4 22
16. 1_{k} 10_{i} 2_{+} 0 14 0 1^{4} 10_{i} −1 2 1^{2} 1_{i} 7∗ 7∗ 4 22 4 22  4 2 22
D_{4}  2  -49  1^{2} 1_{k} 7 7 2 22 22 4 2 22 4 22

Discriminant class: 1

∞ 2 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  3 5 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. 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
18. 2_{−} 4_{−} 2_{+} 0 -3 0 1^{2} 2_{i} −1 −1 2^{2} 2_{+} 1 V 22 22 22 22 22 22 22
V_{4} 4 1^{2} 2_{i} −1 −1 22 22 22 22 22 22
19. 2_{−} 6_{−} 2_{+} -2 9 -8 1^{2} 3_{i} −2 −2 2^{2} 2_{+} 1 V 22 22  22 22 22 22 22
V_{4}  2  1^{2} 3_{i} −2 −2  22 22 22  22 22 22 22
20. 2_{−} 6_{+} 2_{−} 2 1 0 1^{2} 3_{+} 2  2 1^{2} 1_{i} 7∗ 7∗ 22 22 22 22 22 22  22
V_{4} 2  14  1^{2} 3_{+} 2  2 1^{2} 1_{i} 7∗ 7∗ 22 22 22 22 22 22 22 22
21. 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} 4  49  22 22 22 22 22 22 22
22. 0_{+} 8_{−} 2_{−} 0 -8 0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{2} 1_{k} 7 7 22 22 22 22 22 22 22 22
V_{4}  9  1^{2} 1_{k} 7 7  22 22 22  22 22 22
23. 2_{−} 8_{−} 2_{+} 0 8 0 1^{4} 8_{−} 1 {−2 \atop −1} 2^{2} 2_{+} 1 V  22 22 22 22  22 22
V_{4} 4  9  22 22 22 22 22 22  22
24. 2_{−} 6_{−} 2_{+} 2 2 0 1^{4} 6_{−} 1  1^{3} 2_{+} 1  3 3 3 22 3 3 3 22
A_{4}  2  3 3 22 22 3 3 3 3

Discriminant class: 2

∞  2 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  3 5 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
25. 2_{−} 11_{−} 0_{+} 0 4 0 1^{4} 11_{−} 2  2  4 4 4 4  4  4
C_{4}  2  22 4 22 4 22 4 4 4
26. 0_{+} 11_{+} 0_{+} 0 -4 0 1^{4} 11_{+} 2  2 2 0_{+} ∗ ∗ 4 4 4 4  4 22 4
C_{4}  2  2 0_{+} ∗ ∗  4 22 4  4 4 4
27. 0_{+} 11_{−} 2_{−} 0 -28 0 1^{4} 11_{−} 2  2 1^{2} 1_{i} 7∗ 7∗ 4 4 4 4 22 4  4
C_{4}  98  1^{2} 1_{i} 7∗ 7∗  4  4  4 4 4
28. 2_{−} 11_{+} 2_{−} 0 28 0 1^{4} 11_{+} 2  2 1^{2} 1_{k} 7 7 4 4 4 4 22 4 22 4
C_{4}  4  98  1^{2} 1_{k} 7 7 22 4  4 22 4 4 4
29. 2_{−} 3_{−} 2_{+} -1 0 1 1^{2} 3_{−} 2∗  2∗ 2^{2} 2_{+} 1 V 4 4 2 4 22 4  2
D_{4}  1  2 0_{+} ∗  ∗  22 2 22 2 22 2 4 4
30. 2_{−} 9_{−} 0_{+} 0 -2 0 1^{4} 9_{−} 2 −1 2 0_{+} ∗ ∗ 4 2 4 2 22 4 22 2
D_{4} 2 2 0_{+} ∗ ∗ 22 2 4 22 2 4 2
31. 2_{−} 5_{+} 2_{−} 0 -1 0 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{i} 7∗ 7∗ 4 4 2 4 22 4 22 2
D_{4}  2  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{i} 7∗ 7∗ 22 2 22 2 22 2 4 4

Discriminant class: 2 (continued)

∞  2 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  3 5 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
32. 2_{−} 11_{−} 0_{+} 0 0 0 1^{4} 11_{−} 2 −2 2 0_{+} ∗ ∗ 2 4 2 4 22 2 22 4
D_{4} 2 2 0_{+} ∗ ∗ 22 4 22 2 22 4 2 4
33. 2_{−} 9_{−} 2_{+} 0 14 0 1^{4} 9_{−} 2 −1 2^{2} 2_{+} 1 V 4 2 4 2  4 22 2
D_{4} 2 98 22 2 22 4 22 2 4 2
34. 0_{+} 9_{+} 2_{+} 0 -6 0 1^{4} 9_{+} 2 −1 2^{2} 2_{+} 1 V 2 4 4 4 22 2 22 2
D_{4} 2 22 2 22 4 22 2 2 4
35. 2_{−} 9_{+} 2_{−} 0 6 0 1^{4} 9_{+} 2 −1 1^{2} 1_{k} 7 7 2 4 4 4 22 2 22 2
D_{4} 2 2 1^{2} 1_{k} 7 7 2 22 4 2 2 4
36. 2_{−} 11_{−} 2_{+} 0 8 0 1^{4} 11_{−} 2 −2 2^{2} 2_{+} 1 V 4 2 2 2 22 4 22 4
D_{4} 2 2 4 22 2 4 4 2
37. 2_{−} 11_{−} 2_{+} 0 0 0 1^{4} 11_{−} 2 −2 2^{2} 2_{+} 1 V 2 4 2 4  2 22 4
D_{4} 2 98 22 4 2 22 4 2 4
38. 0_{+} 11_{−} 2_{−} 0 -8 0 1^{4} 11_{−} 2 −2 1^{2} 1_{k} 7 7 4 2 2 2 22 4 22 4
D_{4} 2 1^{2} 1_{k} 7 7 22 4 22 2 22 4 4 2
39. 2_{−} 11_{+} 2_{−} 0 -4 0 1^{4} 11_{+} 2  2 1^{2} 1_{i} 7∗ 7∗ 2 2 4 2 22 2 22 4
D_{4}  8  18  1^{2} 1_{i} 7∗ 7∗ 22 4 22 4 22 4 2 2
40. 2_{−} 11_{−} 2_{+} 0 4 0 1^{4} 11_{−} 2  2 2^{2} 2_{+} 1 V 2 2 4 2 22 2  4
D_{4}  4  18  22 4 22 4 22 4 2 2

Discriminant class: 7

∞  2 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  3 5 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
41. 2_{−} 2_{i} 3_{i} 1 3 4 1^{2} 2_{i} −∗ −∗ 1^{4} 3_{i} 7 7∗ 22 4 2 4 4 22 2 22
D_{4} 2 2 0_{+} ∗ ∗ 22 22 4 2 22 22 4
42. 2_{−} 10_{k} 1_{k} 0 6 0 1^{4} 10_{k} −1  2 1^{2} 1_{k} 7 7 22 4 4 4 2 22 2 22
D_{4} 2 7 22 2 4 22 22 4
43. 0_{+} 10_{k} 1_{i} 0 -6 0 1^{4} 10_{k} −1  2 1^{2} 1_{i} 7∗ 7∗ 22 4 4 4 2 22 2 22
D_{4} 7 2 0_{+} ∗ ∗ 22 22 2 4 22 22 22 4
44. 2_{−} 6_{i} 3_{i} 0 0 0 1^{2} 3_{k} −2∗ −2∗ 1^{4} 3_{i} 7 7∗ 22 4 2 4 4 22 2
D_{4}  2  28  1^{2} 3_{−} 2∗  2∗  22  4 2 22 22 22 4
45. 2_{−} 10_{i} 3_{i} 0 -14 0 1^{4} 10_{i} −1  2 1^{4} 3_{i} 7 7∗ 22 2 4 2 4  2 22
D_{4}  8  63  22 22 4 4 22 22 22 2
46. 2_{−} 10_{i} 3_{i} 0 14 0 1^{4} 10_{i} −1  2 1^{4} 3_{i} 7 7∗  2 4 2 4 22 2 22
D_{4}  8  63  22 22 4 4 22 22  2

Discriminant class: 14

∞  2 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  3 5 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
47. 2_{−} 3_{i} 3_{i} 1 3 -3 1^{2} 3_{i} −2 −2 1^{4} 3_{i} 7 7∗ 4 22 22 22 4 4 2 2
D_{4}  2  22 2 4 22 22 2 4 22
48. 2_{−} 5_{i} 3_{i} 0 7 0 1^{2} 3_{+} 2  2 1^{4} 3_{i} 7 7∗ 4 22  22 4 4 2 2
D_{4}  2  14  1^{2} 2_{i} −1 −1  22 2 4  22 2 4 22
49. 0_{+} 11_{i} 1_{k} 0 -8 0 1^{4} 11_{i} −2 2 1^{2} 1_{k} 7 7 4 22 22 22 2 4 2 4
D_{4}  14  22 4 2 22 22 4 4 22

Discriminant class: 14 (continued)

∞  2 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  3 5 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
50. 2_{−} 11_{i} 1_{i} 0 8 0 1^{4} 11_{i} −2 2 1^{2} 1_{i} 7∗ 7∗ 4 22 22 22 2 4 2 4
D_{4} 2 14 2 0_{+} ∗ ∗ 4 2 22 4 4 22
51. 2_{−} 11_{i} 3_{i} 0 0 0 1^{4} 11_{i} −2 2 1^{4} 3_{i} 7 7∗ 2  22 22 4 2 2 4
D_{4} 16 14 22 4 4 22 22 4 2
52. 2_{−} 11_{i} 3_{i} 0 0 0 1^{4} 11_{i} −2 2 1^{4} 3_{i} 7 7∗ 2 22 22  4 2 2 4
D_{4}  8  56  22 4 4 22 22 4 2 22