Quartic Fields Unramified Outside of {∞, 2, 5}

Discriminant class: −10

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  3 7 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_{i} 3_{+} 0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{+} 5∗ 5∗ 2 22 22  4 22 22 4
D_{4} 2 -40 2 22 2 22 22 22 4
2. 1_{k} 11_{i} 3_{+} 0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{+} 5∗ 5∗ 2 22 22 22 4 22 22 4
D_{4} 2 -10 2 2 22 22 4

Discriminant class: −5

∞ 2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 7 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
3. 1_{k} 4_{k} 3_{−} 0 0 0 2^{2} 4_{k} −∗ ∗ 1^{4} 3_{−} 5 5 22 22 2 4 4 2 22 22
D_{4} -5 2 4 22 22 22 4 2 22
4. 1_{k} 6_{k} 3_{−} 0 0 0 2^{2} 6_{k} −∗ ∗ 1^{4} 3_{−} 5 5 22 22 2 4 4 2 22
D_{4} -20 2 4 22 22 22 4 2

Discriminant class: −2

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 7 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
5. 1_{k} 11_{i} 0_{+} 0 0 0 1^{4} 11_{i} −2  2 4 0_{+} ∗ ∗ 22 2 22 4 22 22 2 4
D_{4}  -2  2 4 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 2 22 4  22 2 4
D_{4}  2  -50  2 4 22 22 2 4 22 4
7. 1_{k} 9_{i} 2_{+} 0 -4 -4 1^{4} 9_{i} −2 −∗ 1^{3} 2_{+} ∗  3 4 3 4 3 3 2 4
S_{4} -3 2 4 3 22 4 2 22 4
8. 1_{k} 11_{i} 2_{+} 0 -4 16 1^{4} 11_{i} −2 −2∗ 1^{3} 2_{+} ∗  3 2 3 4 3 3 4 4
S_{4}  -18  4 2 3 22 4 4 22 2
9. 1_{k} 11_{i} 2_{+} 0 4 -8 1^{4} 11_{i} −2  2 1^{3} 2_{+} ∗  3 4 3 2 3 3 4 2
S_{4}  2  2  4 4 3 22 2 4  4

Discriminant class: −1

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  3 7 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
10. 1_{k} 4_{k} 2_{−} 0 -1 0 2^{2} 4_{k} −1 ∗ 2^{2} 2_{−} 1 V 4 4 2 22 22 2 4
D_{4} -1 2 22 22 4 4 22 2 22
11. 1_{k} 10_{i} 0_{+} 0 -2 0 1^{4} 10_{i} −1 2 2 0_{+} ∗ ∗ 4 2 4 22 22 4 2 22
D_{4} -1 2 0_{+} ∗ ∗ 2 22 4 2 22 4 22
12. 1_{k} 6_{i} 2_{+} 0 2 0 2^{2} 6_{i} −1 ∗ 1^{2} 1_{+} 5 5 4 4 2 22 22 2 4 22
D_{4} -4 1^{2} 1_{+} 5 5 2 22 22 4 4 22 2
13. 1_{k} 10_{k} 2_{−} 0 10 0 1^{4} 10_{k} −1 2 2^{2} 2_{−} 1 V 4 2 4 22  4 2 22
D_{4}  2  -25  2 22  4 2 22 4 22

Discriminant class: −1 (continued)

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  3 7 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
14. 1_{k} 10_{i} 2_{+} 0 -2 0 1^{4} 10_{i} −1 2∗ 1^{2} 1_{−} 5∗ 5∗ 2 4 4 22 22 4 4 22
D_{4} 2 -9 1^{2} 1_{−} 5∗ 5∗ 2 22 2 4 22 4 22
15. 1_{k} 10_{k} 2_{−} 0 6 0 1^{4} 10_{k} −1 2∗ 2^{2} 2_{−} 1 V 2 4 4  22 4 4 22
D_{4}  2  -1  2 22 22 2 4  4 22

Discriminant class: 1

∞ 2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  3 7 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
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 3 0 2^{2} 4_{−} 1 {−1 \atop −∗} 1^{2} 1_{+} 5 5 22 22 22 22 22 22 22
V_{4}  1  1^{2} 1_{+} 5 5 22 22  22 22 22 22
18. 2_{−} 6_{+} 2_{−} 0 6 0 2^{2} 6_{+} 1 {−2 \atop −2∗} 2^{2} 2_{−} 1 V 22 22  22 22  22 22
V_{4}  4  22 22  22 22 22  22
19. 0_{+} 6_{−} 2_{−} 0 -6 0 2^{2} 6_{−} 1  {∗ \atop 2} 2^{2} 2_{−} 1 V 22 22 22 22 22 22 22 22
V_{4} 4 22 22 22 22 22 22
20. 2_{−} 8_{−} 2_{+} 0 0 0 1^{4} 8_{−} 1 {−2∗ \atop −1} 1^{2} 1_{−} 5∗ 5∗ 22 22 22  22 22 22 22
V_{4} 2 25 1^{2} 1_{−} 5∗ 5∗ 22 22 22 22 22
21. 2_{−} 8_{+} 2_{−} 0 -4 0 1^{4} 8_{+} 1 {−2 \atop −∗} 2^{2} 2_{−} 1 V  22 22 22 22 22 22 22
V_{4} 2  9  22 22  22 22 22 22
22. 2_{−} 8_{+} 2_{−} 0 4 0 1^{4} 8_{+} 1 {−2∗ \atop −∗} 2^{2} 2_{−} 1 V 22  22 22 22 22  22
V_{4} 2  9  22 22  22  22 22 22

Discriminant class: 2

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  3 7 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
23. 2_{−} 11_{−} 0_{+} 0 4 0 1^{4} 11_{−} 2 2 4 0_{+} ∗ ∗ 4  4 4  4  4
C_{4}  2  22 4 22 4 22 4 4 4
24. 0_{+} 11_{+} 0_{+} 0 -4 0 1^{4} 11_{+} 2 2 4 0_{+} ∗ ∗ 4 22 4 4  4 22 4
C_{4}  2  4 22 4  4 4 4
25. 0_{+} 11_{+} 2_{+} 0 -20 0 1^{4} 11_{+} 2 2 2^{2} 2_{+} ∗ ∗ 4  4 4 22 4  4
C_{4}  2  50  4 22 4 22 4 4 4
26. 2_{−} 11_{−} 2_{+} 0 20 0 1^{4} 11_{−} 2 2 2^{2} 2_{+} ∗ ∗ 4 22 4 4 22 4 22 4
C_{4} 2 50 22 4 22 4 4 4 4
27. 2_{−} 9_{−} 0_{+} 0 -2 0 1^{4} 9_{−} 2 −1 2 0_{+} ∗ ∗ 4 22 4 2 22 4 22 2
D_{4} 2 22 2 4 22 2 4 2
28. 2_{−} 11_{−} 0_{+} 0 0 0 1^{4} 11_{−} 2 −2 4 0_{+} ∗ ∗ 2 22 2 4 22 2 22 4
D_{4} 2 22 4 22 2 22 4 2 4
29. 2_{−} 9_{+} 2_{−} 0 10 0 1^{4} 9_{+} 2 −1 1^{2} 1_{+} 5 5 4 22 4 2  4 22 2
D_{4} 2 50 1^{2} 1_{−} 5∗ 5∗ 22 2 4 22 2 4 2
30. 2_{−} 11_{−} 2_{+} 0 0 0 1^{4} 11_{−} 2 −2 2^{2} 2_{+} ∗ ∗ 2 22 2 4  2 22 4
D_{4}  2  50  22 4 22 2 22 4 2 4

Discriminant class: 5

∞ 2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  3 7 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
31. 2_{−} 0_{+} 3_{−} -1 1 -1 4 0_{+} ∗ ∗ 1^{4} 3_{−} 5 5 4 4  4 4 22 4 22
C_{4} 1 4 4 4 4 22
32. 0_{+} 4_{−} 3_{−} 0 -5 0 2^{2} 4_{−} ∗ ∗ 1^{4} 3_{−} 5 5 4 4 22 4 4  4 22
C_{4} 5 22 4 4 4 4
33. 0_{+} 6_{−} 3_{−} 0 -10 0 2^{2} 6_{−} ∗ ∗ 1^{4} 3_{−} 5 5 4 4  4 4 22 4
C_{4} 20 22 4 4 4 4 22 22
34. 2_{−} 6_{+} 3_{−} 0 10 0 2^{2} 6_{+} ∗ ∗ 1^{4} 3_{−} 5 5 4 4 22 4 4  4
C_{4} 2 20 4 4 4 4 22
35. 2_{−} 6_{+} 1_{−} 2 4 2 1^{4} 6_{+} ∗ −1 1^{2} 1_{−} 5∗ 5∗ 4 4 22 2 2 22 4
D_{4} 1 2 0_{+} ∗ ∗ 22 2 22 4 4 2 22 22
36. 2_{−} 8_{−} 1_{+} 0 -4 0 1^{4} 8_{−} ∗ −1 1^{2} 1_{+} 5 5 4 4 22 2 2 22 4 22
D_{4} 5 22 2 22 4 4 2 22
37. 2_{−} 6_{+} 3_{−} -2 4 2 1^{4} 6_{+} ∗ −∗ 1^{4} 3_{−} 5 5 2 2 22 4 4 22 2 22
D_{4} 4 1 22 4 22 2 2 4 22 22
38. 2_{−} 8_{+} 3_{−} 0 0 0 1^{4} 8_{+} ∗ −∗ 1^{4} 3_{−} 5 5 2 2 22 4 4 22 2
D_{4} 4  5  22 4 22 2 2 4 22

Discriminant class: 10

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  3 7 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
39. 0_{+} 11_{+} 3_{+} 0 -20 0 1^{4} 11_{+} 2∗ 2∗ 1^{4} 3_{+} 5∗ 5∗  4 4 22 4 4 4 4
C_{4} 2 10 22 22 22 22 4 4 4
40. 0_{+} 11_{+} 3_{+} 0 -20 0 1^{4} 11_{+} 2∗ 2∗ 1^{4} 3_{+} 5∗ 5∗ 22 4 4  4 4 4 4
C_{4} 2 90 22 22 4 22 4 4
41. 2_{−} 11_{−} 3_{+} 0 20 0 1^{4} 11_{−} 2∗ 2∗ 1^{4} 3_{+} 5∗ 5∗ 22 4 4 22 4 4 4 4
C_{4}  4  10  22 22  4  4 4
42. 2_{−} 11_{−} 3_{+} 0 20 0 1^{4} 11_{−} 2∗ 2∗ 1^{4} 3_{+} 5∗ 5∗  4 4  4 4 4 4
C_{4} 20  90  22 22 4 22 4 4
43. 2_{−} 9_{+} 1_{−} 0 6 0 1^{4} 9_{+} 2∗ −1 1^{2} 1_{−} 5∗ 5∗ 22 4 4 22 2 4 4 2
D_{4} 10 22 22 22 4 22 4 2
44. 2_{−} 9_{−} 1_{+} 0 2 0 1^{4} 9_{−} 2∗ −1 1^{2} 1_{+} 5 5 22 4 4  2 4 4 2
D_{4} 10 2 0_{+} ∗ ∗ 22 22 22 22 4 4 2
45. 2_{−} 11_{−} 3_{+} 0 0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{+} 5∗ 5∗ 22 2 2  4 2 2 4
D_{4} 4 10 22 22 22 2 22 2 4
46. 2_{−} 11_{−} 3_{+} 0 0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{+} 5∗ 5∗ 22 2 2 22 4 2 2 4
D_{4}  4  40  22  22 2  2 4