Quartic Fields Ramifying Above {2, 23}

Discriminant class: −46

∞  2 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 23  3 5 7 11 13 17 19 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} 23∗ 23∗ 4 22 2 22 4 2 22 4
D_{4} 2 -46 22 22 22 22 4 22
2. 1_{k} 11_{−} 1_{k}  0 4 0 1^{4} 11_{−} 2 2 1^{2} 1_{k} 23 23 4 22 2 22 4 2 22 4
D_{4} -46 2 0_{+} ∗ ∗ 22 22 22 22 4 22
3. 1_{k} 9_{−} 3_{k}  0 -46 0 1^{4} 9_{−} 2 −1 1^{4} 3_{k} 23∗ 23 4 22 2 22 2 4 22 2
D_{4}  2  -46  22 22 22 22 22 22 4 22
4. 1_{k} 9_{−} 3_{k}  0 46 0 1^{4} 9_{−} 2 −1 1^{4} 3_{k} 23∗ 23 4 22 2  2 4  2
D_{4} 10  -46  22 22 22  22 22 4 22
5. 1_{k} 11_{−} 3_{k}  0 0 0 1^{4} 11_{−} 2 −2 1^{4} 3_{k} 23∗ 23 2 22 2 22 4 4 22 4
D_{4}  4  -184  22 22 22 22 22 22 2
6. 1_{k} 11_{−} 3_{k}  0 0 0 1^{4} 11_{−} 2 −2 1^{4} 3_{k} 23∗ 23 2  2 22 4 4 22 4
D_{4}  2  -46  22  22 22 22  2 22

Discriminant class: −23

∞ 2 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 23  3 5 7 11 13 17 19 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 3 -2 2^{2} 6_{−} 1  {∗ \atop 2} 1^{2} 1_{k} 23 23 22 4 2 4 22 2 4 22
D_{4} -1 2 0_{+} ∗ ∗ 22 4 22 4 4 22 4
8. 1_{k} 8_{+} 1_{i}  0 -4 -8 1^{4} 8_{+} 1 {−2∗ \atop −∗} 1^{2} 1_{i} 23∗ 23∗ 22 4 2 4 22 2 4 22
D_{4} -7 4 22 4 22 4 22 4
9. 1_{k} 4_{−} 3_{k} -2 -10 34 2^{2} 4_{−} 1 {−1 \atop −∗} 1^{4} 3_{k} 23∗ 23 22 4 2 2  4 2 22
D_{4} 13 22 4 2 22 4 22 4
10. 1_{k} 6_{−} 3_{k}  2 13 12 1^{2} 3_{i} −2 −2 1^{4} 3_{k} 23∗ 23 22 2 2 4 22 4 4 22
D_{4}  -10  1^{2} 3_{i} −2 −2  22 2 22 4 22 2  2
11. 1_{k} 8_{−} 3_{k}  0 0  0 1^{4} 8_{−} 1 {−2∗ \atop −1} 1^{4} 3_{k} 23∗ 23 22 4 2 2 22 4 2
D_{4} 2  -23  22 4  2 22 4 22 4
12. 1_{k} 8_{−} 3_{k}  0 0 -184 1^{4} 8_{−} 1 {−2 \atop −1} 1^{4} 3_{k} 23∗ 23  2 2 4 22 4 4 22
D_{4} 2  345  22 2 22 4 22 2 22 2
13. 1_{k} 6_{−} 1_{k}  2 2  2 1^{4} 6_{−} 1  1^{2} 1_{k} 23 23 3 4 4 4 3 4 2 3
S_{4}  -1  2 0_{+} ∗  ∗ 3 4 3 2 3 2 22 2
14. 1_{k} 6_{−} 3_{k}  2 -10 -34 1^{4} 6_{−} 1  1^{4} 3_{k} 23∗ 23 3 2 4 2 3 4 4 3
S_{4}  -33  3 2 3 2 3 4 22 4
15. 1_{k} 6_{−} 3_{k}  2 -10 12 1^{4} 6_{−} 1  1^{4} 3_{k} 23∗ 23 3 4 2 4 3 2 4 3
S_{4}  82  3 4 3 2 3 4  4

Discriminant class: −2

∞  2 23  a_{1} a_{2} a_{3}  Over p = 2  Over p = 23  3 5 7 11 13 17 19 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. 1_{k} 11_{i} 2_{+}  0 0 0 1^{4} 11_{i} −2  2 1^{2} 1_{i} 23∗ 23∗ 22 4 2 22 4  22 4
D_{4}  2  -1058  1^{2} 1_{k} 23 23 2 4 22 22 2 4 22 4

Discriminant class: −2 (continued)

∞  2 23  a_{1} a_{2} a_{3}  Over p = 2  Over p = 23  3 5 7 11 13 17 19 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. 1_{k} 9_{i} 2_{+}  0 -6 -8 1^{4} 9_{i} −2 −∗ 1^{3} 2_{+} ∗  3 2 4 22 2 3 3 2
S_{4}  -26  4 2 22 3 4 4 3 4
18. 1_{k} 11_{i} 2_{+}  0 -8 -32 1^{4} 11_{i} −2 −2∗ 1^{3} 2_{+} ∗  3 4 2 22 2 3 3 2
S_{4}  56  2 4 22 3 2 4 3 4
19. 1_{k} 11_{i} 2_{+}  0 -4 40 1^{4} 11_{i} −2  2 1^{3} 2_{+} ∗  3 4 4 22 2 3 3 2
S_{4}  2  -78  4 4  3 4 2 3 2

Discriminant class: −1

∞  2 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 23  3 5 7 11 13 17 19 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
20. 1_{k} 10_{i} 2_{+}  0 46 0 1^{4} 10_{i} −1 2 1^{2} 1_{k} 23 23 4 22 2 4 22  4 22
D_{4}  2  -529  1^{2} 1_{i} 23∗ 23∗ 2 22  4 2 22 4 22
21. 1_{k} 2_{i} 2_{+} -1 1 0 1^{2} 2_{i} −∗ −∗ 1^{3} 2_{+} ∗  4 3 4 4 3 22 4 3
S_{4}  -2  2 0_{+} ∗  ∗  2 22 3 2 4 3 4 3
22. 1_{k} 6_{i} 2_{+}  2 7 -2 2^{2} 6_{i} −1 ∗ 1^{3} 2_{+} ∗  4 3 2 4 3 22 2 3
S_{4}  -7  4  3 4 4 3 2 3
23. 1_{k} 6_{i} 2_{+}  2 -5 10 2^{2} 6_{i} −1 ∗ 1^{3} 2_{+} ∗  2 3 4 2 3 22 4 3
S_{4}  1  4 22 3 4 2 3 4 3

Discriminant class: 1

∞ 2 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 23  3 5 7 11 13 17 19 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
24. 2_{−} 4_{−} 2_{+} 0 -11 0 1^{2} 2_{i} −1 −1 2^{2} 2_{+} 1  V 22 22 22 22  22 22
V_{4} 3  36  1^{2} 2_{i} −1 −1  22 22  22 22 22 22 22
25. 2_{−} 6_{−} 2_{+} -2 17 -16 1^{2} 3_{i} −2 −2 2^{2} 2_{+} 1  V  22 22 22 22 22 22 22
V_{4} 3  18  1^{2} 3_{i} −2 −2  22 22  22 22 22  22
26. 2_{−} 6_{+} 2_{−} 2 9 8 1^{2} 3_{+} 2  2 1^{2} 1_{i} 23∗ 23∗ 22 22 22 22 22 22 22 22
V_{4} 6  62  1^{2} 3_{+} 2  2 1^{2} 1_{i} 23∗ 23∗  22  22  22 22 22
27. 2_{−} 8_{−} 2_{+} 0 0 0 1^{4} 8_{−} 1 {−2 \atop −1} 2^{2} 2_{+} 1  V 22  22 22 22 22 22 22
V_{4} 4  529  22  22 22  22
28. 2_{−} 8_{−} 2_{+} 0 24 0 1^{4} 8_{−} 1 {−2 \atop −1} 2^{2} 2_{+} 1  V 22 22 22  22 22  22
V_{4} 4  121  22 22  22 22 22 22
29. 0_{+} 8_{−} 2_{−} 0 -24 0 1^{4} 8_{−} 1 {−2 \atop −1} 1^{2} 1_{k} 23 23 22 22  22 22 22 22 22
V_{4}  121  1^{2} 1_{k} 23 23 22 22  22 22 22 22 22

Discriminant class: 2

∞  2 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 23  3 5 7 11 13 17 19 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
30. 2_{−} 11_{+} 2_{−}  0 92 0 1^{4} 11_{+} 2  2 1^{2} 1_{k} 23 23 4 4  4 4 22 4 4
C_{4} 20  1058  1^{2} 1_{k} 23 23  4 22 4  4 4 4

Discriminant class: 2 (continued)

∞  2 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 23  3 5 7 11 13 17 19 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. 0_{+} 11_{−} 2_{−}  0 -92 0 1^{4} 11_{−} 2  2 1^{2} 1_{i} 23∗ 23∗ 4 4 22 4 4 22 4 4
C_{4}  1058  1^{2} 1_{i} 23∗ 23∗ 22 4 22 4 22 4 4 4
32. 2_{−} 3_{−} 2_{+}  -1 4 1 1^{2} 3_{−} 2∗  2∗ 2^{2} 2_{+} 1  V 2 4 22 4 2 22 4 2
D_{4}  3  1  2 0_{+} ∗  ∗  22 4 22 4  4 2 4
33. 2_{−} 5_{+} 2_{−}  0 -3 0 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{i} 23∗ 23∗ 2 4 22 4 2 22 4 2
D_{4} 3 8 1^{2} 2_{i} −∗ −∗ 1^{2} 1_{i} 23∗ 23∗ 4 22 4 22 4 2 4
34. 2_{−} 9_{+} 2_{−}  0 10 0 1^{4} 9_{+} 2 −1 1^{2} 1_{k} 23 23 4 4 22 2 2 22 2 2
D_{4}  2  2  1^{2} 1_{k} 23 23 22 4 22 2 22 4 4 4
35. 2_{−} 9_{−} 2_{+}  0 46 0 1^{4} 9_{−} 2 −1 2^{2} 2_{+} 1  V 4 2 22 4 2  4 2
D_{4}  2  1058  22 2  4 22 2 4 2
36. 0_{+} 9_{+} 2_{+}  0 -10 0 1^{4} 9_{+} 2 −1 2^{2} 2_{+} 1  V 4 4  2 2 22 2 2
D_{4}  2  22 4 22 2 22 4 4 4
37. 2_{−} 11_{−} 2_{+}  0 16 0 1^{4} 11_{−} 2 −2 2^{2} 2_{+} 1  V 2 2  4 4 22 4 4
D_{4}  6  18  22 2  4 22 2 2 2
38. 0_{+} 11_{−} 2_{−}  0 -16 0 1^{4} 11_{−} 2 −2 1^{2} 1_{k} 23 23 2 2 22 4 4 22 4 4
D_{4}  18  1^{2} 1_{k} 23 23 22 2  4 22 2 2 2
39. 2_{−} 11_{−} 2_{+}  0 4 0 1^{4} 11_{−} 2  2 2^{2} 2_{+} 1  V 4 2 22 2 4 22 2 4
D_{4}  4  50  2  2 22 2 4 2
40. 2_{−} 11_{+} 2_{−}  0 -4 0 1^{4} 11_{+} 2  2 1^{2} 1_{i} 23∗ 23∗ 4 2 22 2 4 22 2 4
D_{4}  8  50  1^{2} 1_{i} 23∗ 23∗ 22 2  2  2 4 2
41. 2_{−} 11_{−} 2_{+}  0 0 0 1^{4} 11_{−} 2 −2 2^{2} 2_{+} 1  V 2 4 22 2 4  2 4
D_{4}  2  1058  22 4 22 2 22 4 2 4

Discriminant class: 23

∞  2 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 23  3 5 7 11 13 17 19 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
42. 2_{−} 10_{k} 1_{k}  0 10 0 1^{4} 10_{k} −1  2 1^{2} 1_{k} 23 23 4 4 22 22 22 2 22 22
D_{4}  2  23  2 4 22 22 2 4 4 4
43. 0_{+} 10_{k} 1_{i}  0 -10 0 1^{4} 10_{k} −1  2 1^{2} 1_{i} 23∗ 23∗ 4 4  22 22 2 22 22
D_{4}  23  2 0_{+} ∗  ∗ 2 4 22 22 2 4 4 4
44. 2_{−} 2_{i} 3_{i}  0 0 0 1^{2} 2_{i} −∗ −∗ 1^{4} 3_{i} 23 23∗ 2 4 22 22  4 22 22
D_{4}  3  23  2 0_{+} ∗  ∗  2 4  22 2 4 2 4
45. 2_{−} 6_{i} 3_{i}  0 0 0 1^{2} 3_{k} −2∗ −2∗ 1^{4} 3_{i} 23 23∗ 2 4 22 22 22 4 22
D_{4}  6  92  1^{2} 3_{−} 2∗  2∗  2 4  22 2 4 2 4
46. 2_{−} 10_{i} 3_{i}  0 46 0 1^{4} 10_{i} −1  2 1^{4} 3_{i} 23 23∗ 4 2 22 22 22 4 22 22
D_{4}  8  575  2 2 22 22 2 2 4 2
47. 2_{−} 10_{i} 3_{i}  0 -46 0 1^{4} 10_{i} −1  2 1^{4} 3_{i} 23 23∗ 4 2 22  22 4  22
D_{4} 40  575  2 2 22  2 2 4 2

Discriminant class: 46

∞  2 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 23  3 5 7 11 13 17 19 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
48. 2_{−} 11_{i} 1_{i}  0 16 0 1^{4} 11_{i} −2 2 1^{2} 1_{i} 23∗ 23∗ 22 22  4 4 2 4 4
D_{4}  6  46  2 0_{+} ∗  ∗ 2 22  4 2 22 22 22

Discriminant class: 46 (continued)

∞  2 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 23  3 5 7 11 13 17 19 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
49. 0_{+} 11_{i} 1_{k}  0 -16 0 1^{4} 11_{i} −2 2 1^{2} 1_{k} 23 23 22 22 22 4 4 2 4 4
D_{4} 46 2 22 4 2 22 22 22
50. 2_{−} 3_{i} 3_{i} -1 9 -13 1^{2} 3_{i} −2 −2 1^{4} 3_{i} 23 23∗ 22 22 22 4 2 4 4 2
D_{4}  3  8  2 22 22 4 2 22  22
51. 2_{−} 5_{i} 3_{i}  0 -23 0 1^{2} 3_{+} 2  2 1^{4} 3_{i} 23 23∗  22 22 4 2 4 4 2
D_{4}  6  184  1^{2} 2_{i} −1 −1  2 22 22 4 2 22 22 22
52. 2_{−} 11_{i} 3_{i}  0 0 0 1^{4} 11_{i} −2 2 1^{4} 3_{i} 23 23∗ 22  22 2 4 4 2 4
D_{4}  8  184  2  22 2 2  22 22
53. 2_{−} 11_{i} 3_{i}  0 0 0 1^{4} 11_{i} −2 2 1^{4} 3_{i} 23 23∗ 22 22 22 2 4 4 2 4
D_{4} 16  46  2 22 22 2 2 22 22