Quartic Fields Unramified Outside of {∞, 2, 11}

Discriminant class: −22

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 5 7 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_{−} 3_{k} 0 0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{k} 11 11∗ 2 4 2 22 4 22 22
D_{4} -88 22 4 4 22 22 4 2 22
2. 1_{k} 11_{−} 3_{k} 0 0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{k} 11 11∗ 2 4 2  4 22 22 22
D_{4} -22 22 4 4 22 22 4 2

Discriminant class: −11

∞ 2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 5 7 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} 6_{+} 3_{i}  2 -4 -16 1^{4} 6_{+} ∗ −∗ 1^{4} 3_{i} 11∗ 11 22 22 2 4 4 2 22 4
D_{4}  -2  22  4 2 22  22 4
4. 1_{k} 8_{+} 3_{i}  0 0 0 1^{4} 8_{+} ∗ −∗ 1^{4} 3_{i} 11∗ 11 22  2 4 4 2 22 4
D_{4}  -11  22 22 4 2 22 22 22 4
5. 1_{k} 8_{−} 1_{k} 0 -2 -4 1^{4} 8_{−} ∗  1^{2} 1_{k} 11 11 3 3 4 4 4 4 3 2
S_{4}  -1  2 0_{+} ∗ ∗ 3 3 4 2 22 22 3 2
6. 1_{k} 4_{+} 3_{i}  2 -4 -16 1^{4} 4_{+} ∗  1^{4} 3_{i} 11∗ 11 3 3 4 2 4 2 3 2
S_{4}  -13  3 3 2 4 22 22 3 2
7. 1_{k} 8_{+} 3_{i}  0 0 -44 1^{4} 8_{+} ∗  1^{4} 3_{i} 11∗ 11 3 3 2 4 2 4 3 2
S_{4}  22  3 3 4 4  22 3 2

Discriminant class: −2

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 5 7 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
8. 1_{k} 11_{i} 0_{+}  0 0 0 1^{4} 11_{i} −2  2 2 0_{+} ∗  ∗ 22 4 2 4 22 22 2 4
D_{4}  -2  2 0_{+} ∗  ∗ 2 4 22 22 2 4 22 4
9. 1_{k} 9_{i} 2_{+}  0 6 0 1^{4} 9_{i} −2 −∗ 2^{2} 2_{+} 1  V 22 2 2 4 22  4 4
D_{4}  -2  4 2 22 22 4 2 22 4
10. 1_{k} 9_{k} 2_{−}  0 -6 0 1^{4} 9_{k} −2 −∗ 1^{2} 1_{k} 11 11 22 2 2 4 22 22 4 4
D_{4}  -2  1^{2} 1_{k} 11 11 4 2 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 4  22 2 4
D_{4}  -242  2 4  22 2 4 22 4
12. 1_{k} 11_{i} 2_{+}  0 -4 0 1^{4} 11_{i} −2 −2∗ 2^{2} 2_{+} 1  V  4 2 2 22 22 4 2
D_{4}  -18  4 4 22 22 4 4 22 2
13. 1_{k} 11_{k} 2_{−}  0 4 0 1^{4} 11_{k} −2 −2∗ 1^{2} 1_{i} 11∗ 11∗ 22 4 2 2 22 22 4 2
D_{4}  -18  1^{2} 1_{i} 11∗ 11∗ 4 4 22 22 4 4  2

Discriminant class: −1

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11 3 5 7 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} 0_{+}  0 -2 0 1^{4} 10_{i} −1  2 4 0_{+} ∗ ∗ 4 22 2 22 22 4 2 22
D_{4}  -1  2 22  4 2 22 4 22
15. 1_{k} 10_{k} 2_{−}  0 -22 0 1^{4} 10_{k} −1  2 2^{2} 2_{−} ∗ ∗ 4 22 2 22  4 2 22
D_{4}  -121  2 22 22 4 2 22 4 22
16. 1_{k} 6_{i} 2_{+}  2 -3 0 1^{2} 3_{k} −2∗ −2∗ 1^{3} 2_{+} ∗  4 3 4 3 3 4 4 3
S_{4}  2  1^{2} 3_{−} 2∗  2∗  4 3 3 2 4 3 2 22
17. 1_{k} 10_{i} 2_{+}  0 -8 -16 1^{4} 10_{i} −1 2∗ 1^{3} 2_{+} ∗  2 3 4 3 3 4 2 3
S_{4}  2  24  4 3 3 4 2 3 4 22
18. 1_{k} 10_{i} 2_{+}  0 -6 -8 1^{4} 10_{i} −1 2∗ 1^{3} 2_{+} ∗  4 3 2 3 3 2 4 3
S_{4}  2  -25  2 3 3 4 4 3 4

Discriminant class: 1

∞ 2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 5 7 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
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 -5 0 2^{2} 4_{−} 1 {−1 \atop −∗} 2^{2} 2_{+} 1  V 22  22 22 22 22 22 22
V_{4}  9  22  22 22 22  22 22
21. 2_{−} 6_{+} 2_{−} -2 11 -10 2^{2} 6_{+} 1 {−2 \atop −2∗} 1^{2} 1_{i} 11∗ 11∗  22 22 22 22 22 22 22
V_{4}  3  1^{2} 1_{i} 11∗ 11∗ 22 22 22 22 22 22  22
22. 2_{−} 6_{−} 2_{+} 2 3 2 2^{2} 6_{−} 1  {∗ \atop 2} 2^{2} 2_{+} 1  V 22 22 22 22 22 22  22
V_{4}  23  22 22 22  22 22 22
23. 0_{+} 8_{+} 2_{+} 0 -12 0 1^{4} 8_{+} 1 {−2∗ \atop −∗} 2^{2} 2_{+} 1  V 22 22  22 22 22 22 22
V_{4}  25  22 22 22 22 22 22 22 22
24. 2_{−} 8_{+} 2_{−} 0 12 0 1^{4} 8_{+} 1 {−2 \atop −∗} 1^{2} 1_{k} 11 11 22 22 22 22 22  22 22
V_{4} 2  25  1^{2} 1_{k} 11 11 22 22 22  22 22 22 22
25. 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} 2  121  22 22 22 22 22 22 22

Discriminant class: 2

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 5 7 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
26. 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
27. 0_{+} 11_{+} 0_{+} 0 -4 0 1^{4} 11_{+} 2 2 4 0_{+} ∗  ∗ 4 4 22 4  4 22 4
C_{4}  2  4 22 4  4 4 4
28. 0_{+} 11_{−} 2_{−} 0 -44 0 1^{4} 11_{−} 2 2 2^{2} 2_{−} ∗  ∗ 4 4 22 4 22 4  4
C_{4}  242  22 4  4 22 4 4 4
29. 2_{−} 11_{+} 2_{−} 0 44 0 1^{4} 11_{+} 2 2 2^{2} 2_{−} ∗  ∗ 4 4  4 22 4 22 4
C_{4} 10  242  4  4  4 4 4
30. 2_{−} 9_{−} 0_{+} 0 -2 0 1^{4} 9_{−} 2 −1 4 0_{+} ∗  ∗ 4 2 22 2 22 4 22 2
D_{4}  2  22 2  4 22 2 4 2

Discriminant class: 2 (continued)

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 5 7 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_{−} 11_{−} 0_{+} 0 0 0 1^{4} 11_{−} 2 −2 2 0_{+} ∗  ∗ 2 4 22 4 22 2 22 4
D_{4}  2  22 4 22 2 22 4 2 4
32. 2_{−} 9_{+} 2_{−} 0 22 0 1^{4} 9_{+} 2 −1 2^{2} 2_{−} ∗  ∗ 4 2 22 2  4 22 2
D_{4}  2  242  22 2 22 4 22 2 4 2
33. 2_{−} 11_{−} 2_{+} 0 0 0 1^{4} 11_{−} 2 −2 1^{2} 1_{i} 11∗ 11∗ 2 4 22 4  2 22 4
D_{4}  2  242  1^{2} 1_{k} 11 11 22 4  2 22 4 2 4

Discriminant class: 11

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 5 7 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
34. 2_{−} 10_{i} 1_{i} 0 -6 0 1^{4} 10_{i} −∗ −2 1^{2} 1_{i} 11∗ 11∗ 2 22 22 4 2  4 4
D_{4} 11 2 0_{+} ∗ ∗ 4 22 2 22 4 22 2 4
35. 2_{−} 10_{k} 1_{k} 0 6 0 1^{4} 10_{k} −∗ −2 1^{2} 1_{k} 11 11 2 22 22 4 2 22 4 4
D_{4} 11 4 22 2 4 22 2 4
36. 2_{−} 4_{k} 3_{k} 0 0 0 2^{2} 4_{k} −∗  ∗ 1^{4} 3_{k} 11 11∗ 2 22 22 4 4 22 2 4
D_{4}  11  2  4 22 2  2 4
37. 2_{−} 6_{k} 3_{k} 0 0 0 2^{2} 6_{k} −∗  ∗ 1^{4} 3_{k} 11 11∗ 2  22 4 4 22 2 4
D_{4}  44  2 22 4 22 2 22 2 4
38. 2_{−} 10_{k} 3_{k} 0 22 0 1^{4} 10_{k} −∗ −2∗ 1^{4} 3_{k} 11 11∗ 2 22  2 4 22 4 2
D_{4} 10 99 4 22 4 22 4 22 2 2
39. 0_{+} 10_{i} 3_{k} 0 -22 0 1^{4} 10_{i} −∗ −2∗ 1^{4} 3_{k} 11 11∗ 2 22 22 2 4 22 4 2
D_{4}  99  4 22 4 22 4 22 2 2

Discriminant class: 22

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 5 7 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
40. 2_{−} 11_{i} 1_{i} 0 -4 0 1^{4} 11_{i} −2∗ −2 1^{2} 1_{i} 11∗ 11∗ 22 4 22 22 2 2 4 22
D_{4} 22 4 4 2 2 4 4 22
41. 2_{−} 11_{k} 1_{k} 0 4 0 1^{4} 11_{k} −2∗ −2 1^{2} 1_{k} 11 11  4 22 22 2 2 4 22
D_{4} 22 2 0_{+} ∗ ∗ 4 4 2 2 4 4 22 22
42. 2_{−} 9_{i} 3_{i} 0 22 0 1^{4} 9_{i} −2∗ −∗ 1^{4} 3_{i} 11∗ 11 22 2  22 4 2 4 22
D_{4} 10 22 4 2 4 2 4 2 22 22
43. 0_{+} 9_{k} 3_{i} 0 -22 0 1^{4} 9_{k} −2∗ −∗ 1^{4} 3_{i} 11∗ 11 22 2 22 22 4 2 4 22
D_{4} 22 4 2 4 2 4 2 22 22
44. 2_{−} 11_{i} 3_{i} 0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{i} 11∗ 11 22 4 22 22 4 2 2
D_{4} 2 22 2 4 4 2 2 4 22 22
45. 2_{−} 11_{i} 3_{i} 0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{i} 11∗ 11 22 4 22  4 2 2 22
D_{4}  2  88  2 4 4 2 2 4 22