Quartic Fields Ramifying Above {2, 19}

Discriminant class: −38

∞  2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  3 5 7 11 13 17 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} 19 19∗ 22 4 22 2  22
D_{4}  -152  2 22 4 2 22  22 4
2. 1_{k} 11_{−} 3_{k}  0 0  0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{k} 19 19∗ 22 4 22 2 22  22 22
D_{4}  -38  2  4 2 22 22 22 4
3. 1_{k} 9_{+} 1_{i}  0 -4 -4 1^{4} 9_{+} 2∗ −1 1^{2} 1_{i} 19∗ 19∗ 3 4 3 2 3 3 3 3
S_{4} 1 2 0_{+} ∗ ∗ 2 22 4 4 22 3 3 2
4. 1_{k} 11_{+} 1_{i}  0 -4 -16 1^{4} 11_{+} 2∗ 2∗ 1^{2} 1_{i} 19∗ 19∗ 3 2 3 4 3 3 3 3
S_{4} 2 -6 2 0_{+} ∗ ∗ 4 22 4 2 22 3 3 2
5. 1_{k} 11_{−} 1_{k}  0 -4 -8 1^{4} 11_{−} 2∗ −2∗ 1^{2} 1_{k} 19 19 3 4 3 4 3 3 3 3
S_{4} -2 4 2 4 22 3 3 2
6. 1_{k} 5_{−} 3_{k}  2 -8 -28 1^{2} 3_{i} −2 −2 1^{4} 3_{k} 19 19∗ 3 4 3 2 3 3 3 3
S_{4} 44 1^{2} 2_{i} −∗ −∗ 4 22 2 4 3 3 4
7. 1_{k} 9_{−} 3_{k}  0 -38 -152 1^{4} 9_{−} 2∗ −1 1^{4} 3_{k} 19 19∗ 3 2 3 2 3 3 3 3
S_{4} 2 -190 4 22 4 2 22 3 3 4
8. 1_{k} 11_{−} 3_{k}  0 0 152 1^{4} 11_{−} 2∗ 2∗ 1^{4} 3_{k} 19 19∗ 3 4 3 4 3 3 3 3
S_{4} 4 -266 2 22 4 4 22 3 3 4
9. 1_{k} 11_{−} 3_{k}  0 0 304 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{k} 19 19∗ 3 2 3 4 3 3 3 3
S_{4} 2 -950 2 22 2 2 22 3 3 4

Discriminant class: −19

∞ 2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  3 5 7 11 13 17 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} 6_{+} 3_{i}  2 -8 -28 1^{4} 6_{+} ∗ −∗ 1^{4} 3_{i} 19∗ 19 2  22 22 4 22 22 4
D_{4}  6  2 4 4 22 22 4 2
11. 1_{k} 8_{+} 3_{i}  0 0 0 1^{4} 8_{+} ∗ −∗ 1^{4} 3_{i} 19∗ 19 2 22 22 22 4 22 22 4
D_{4}  -19  2 4 4 22 22 4 2 22
12. 1_{k} 8_{+} 1_{i}  0 0 -4 1^{4} 8_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 4 3 3 3 2 3 22 4
S_{4}  2  4 4 4 3 3 4 2 3
13. 1_{k} 2_{+} 3_{i}  -1 -2 -6 1^{3} 2_{+} ∗  1^{4} 3_{i} 19∗ 19 4 3 3 3 2 3 22 4
S_{4}  -2  2 4 4 3 3 2 4 3
14. 1_{k} 8_{+} 3_{i}  0 0 76 1^{4} 8_{+} ∗  1^{4} 3_{i} 19∗ 19 2 3 3 3 2 3 22 2
S_{4} 4  -114  4 2 2 3 3 4 4 3

Discriminant class: −2

∞  2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  3 5 7 11 13 17 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} 9_{k} 2_{−}  0 -2 0 1^{4} 9_{k} −2 −∗ 1^{2} 1_{k} 19 19 22 2 4 22 4 22 4 4
D_{4}  -18  1^{2} 1_{k} 19 19 2 4 22 22 4 4  2
16. 1_{k} 9_{i} 2_{+}  0 2 0 1^{4} 9_{i} −2 −∗ 2^{2} 2_{+} 1  V  2 4 22 4 22 4 4
D_{4}  -18  2 4 22 22 4 4 22 2

Discriminant class: −2 (continued)

∞  2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  3 5 7 11 13 17 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. 1_{k} 11_{i} 2_{+}  0 0 0 1^{4} 11_{i} −2  2 2^{2} 2_{+} 1  V 22 4 2 22 4 22 2 4
D_{4}  -722  2 4  22 2 4 22 4
18. 1_{k} 11_{k} 2_{−}  0 -12 0 1^{4} 11_{k} −2 −2∗ 1^{2} 1_{i} 19∗ 19∗ 22 4 4 22 2  4 2
D_{4}  -2  1^{2} 1_{i} 19∗ 19∗ 2 2 22  4 2 22 4
19. 1_{k} 11_{i} 2_{+}  0 12 0 1^{4} 11_{i} −2 −2∗ 2^{2} 2_{+} 1  V 22 4 4  2  4 2
D_{4}  -2  2 2 22 22 4 2 22 4
20. 1_{k} 5_{i} 2_{+}  2 4 2 1^{2} 3_{+} 2  2 1^{3} 2_{+} 1  3 4 4 3 2 22 4 4
S_{4}  -13  1^{2} 2_{i} −1 −1  4 4 3 22 4 2 3 4
21. 1_{k} 11_{i} 2_{+}  0 -4 -8 1^{4} 11_{i} −2  2 1^{3} 2_{+} 1  3 2 2 3 4 22 4 2
S_{4}  2  -14  4 4 3 22 2 4 3 2
22. 1_{k} 11_{i} 2_{+}  0 -20 -56 1^{4} 11_{i} −2  2 1^{3} 2_{+} 1  3 4 4 3 4 22 2 4
S_{4}  2  -46  2 2 3  4 4 3 4

Discriminant class: −1

∞  2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19 3 5 7 11 13 17 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. 1_{k} 10_{k} 2_{−}  0 38 0 1^{4} 10_{k} −1 2 2^{2} 2_{−} ∗ ∗ 4 22 2 4 22 22 2 22
D_{4}  -361  2 22 22 4 2 22 4 22

Discriminant class: 1

∞ 2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  3 5 7 11 13 17 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
24. 2_{−} 4_{−} 2_{+} 0 -9 0 2^{2} 4_{−} 1 {−1 \atop −∗} 2^{2} 2_{+} 1  V 22  22 22 22  22 22
V_{4}  25  22 22 22 22 22 22 22
25. 2_{−} 6_{+} 2_{−} 2 15 14 2^{2} 6_{+} 1 {−2 \atop −2∗} 1^{2} 1_{i} 19∗ 19∗ 22 22 22  22  22 22
V_{4}  11  1^{2} 1_{i} 19∗ 19∗ 22 22 22  22 22 22 22
26. 2_{−} 6_{−} 2_{+} 2 7 6 2^{2} 6_{−} 1  {∗ \atop 2} 2^{2} 2_{+} 1  V 22 22  22 22  22
V_{4} 3  47  22 22 22 22  22 22 22
27. 2_{−} 8_{+} 2_{−} 0 20 0 1^{4} 8_{+} 1 {−2 \atop −∗} 1^{2} 1_{k} 19 19  22 22 22 22  22 22
V_{4} 6  81  1^{2} 1_{k} 19 19 22 22 22 22 22 22  22
28. 0_{+} 8_{+} 2_{+} 0 -20 0 1^{4} 8_{+} 1 {−2∗ \atop −∗} 2^{2} 2_{+} 1  V 22 22 22 22 22  22 22
V_{4}  81  22 22 22 22 22 22 22
29. 2_{−} 8_{−} 2_{+} 0 0 0 1^{4} 8_{−} 1 {−2∗ \atop −1} 2^{2} 2_{+} 1  V 22 22 22 22  22
V_{4} 6  361  22  22 22 22  22 22
30. 2_{−} 6_{−} 2_{+} -2 10 -8 1^{4} 6_{−} 1  1^{3} 2_{+} 1  3 3 22 22 3 3 3 3
A_{4}  2  22 3 3 3 3 3 3

Discriminant class: 2

∞  2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  3 5 7 11 13 17 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_{+} 2_{−} 0 76 0 1^{4} 11_{+} 2 2 2^{2} 2_{−} ∗  ∗ 4 4 22 4 4  22 4
C_{4} 10  722  22 4  4  4 4 4
32. 0_{+} 11_{−} 2_{−} 0 -76 0 1^{4} 11_{−} 2 2 2^{2} 2_{−} ∗  ∗ 4 4  4 4  4
C_{4}  722  4  4 22 4 4 4
33. 2_{−} 9_{+} 2_{−} 0 -38 0 1^{4} 9_{+} 2 −1 2^{2} 2_{−} ∗  ∗ 4 2 22 4 2 22 22 2
D_{4} 2 722 22 2 22 4 22 2 4 2
34. 2_{−} 11_{−} 2_{+} 0 0 0 1^{4} 11_{−} 2 −2 1^{2} 1_{k} 19 19 2 4 22 2 4 22 22 4
D_{4}  2  722  1^{2} 1_{i} 19∗ 19∗ 22 4  2 22 4 2 4

Discriminant class: 19

∞  2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  3 5 7 11 13 17 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
35. 2_{−} 10_{k} 1_{k} 0 2 0 1^{4} 10_{k} −∗ −2 1^{2} 1_{k} 19 19 22 22 4 2 4 22 4 4
D_{4} 19 22 4 2 2 4 4 22
36. 2_{−} 10_{i} 1_{i} 0 -2 0 1^{4} 10_{i} −∗ −2 1^{2} 1_{i} 19∗ 19∗  22 4 2 4 22 4 4
D_{4}  19  2 0_{+} ∗  ∗ 22 4 2 2 4 4 22 22
37. 2_{−} 4_{k} 3_{k} 0 0 0 2^{2} 4_{k} −∗  ∗ 1^{4} 3_{k} 19 19∗ 22  2 2 4 22 2 4
D_{4}  19  22 4 4 2 2 4 22
38. 2_{−} 6_{k} 3_{k} 0 0 0 2^{2} 6_{k} −∗  ∗ 1^{4} 3_{k} 19 19∗ 22 22 2 2 4 22 2 4
D_{4} 76 22 4 4 2 2 4 22 22
39. 2_{−} 10_{k} 3_{k} 0 38 0 1^{4} 10_{k} −∗ −2∗ 1^{4} 3_{k} 19 19∗ 22 22 4 2 2  4 2
D_{4} 14 19 22 2 4 2 4 2 22 22
40. 0_{+} 10_{i} 3_{k} 0 -38 0 1^{4} 10_{i} −∗ −2∗ 1^{4} 3_{k} 19 19∗ 22 22 4 2 2  4 2
D_{4}  19  2 4 2 4 2 22 22

Discriminant class: 38

∞  2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  3 5 7 11 13 17 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_{−} 11_{i} 1_{i}  0 12 0 1^{4} 11_{i} −2∗ −2 1^{2} 1_{i} 19∗ 19∗ 2 4 4 22 22  4 22
D_{4}  38  22 22 2  4 22 2 4
42. 2_{−} 11_{k} 1_{k}  0 -12 0 1^{4} 11_{k} −2∗ −2 1^{2} 1_{k} 19 19 2 4 4  22  4 22
D_{4}  38  2 0_{+} ∗  ∗ 22 22 2 22 4 22 2 4
43. 2_{−} 9_{i} 3_{i}  0 38 0 1^{4} 9_{i} −2∗ −∗ 1^{4} 3_{i} 19∗ 19 2 2 4 22 22  4 22
D_{4} 14  342  22 22 4 22 4 22 2 2
44. 0_{+} 9_{k} 3_{i}  0 -38 0 1^{4} 9_{k} −2∗ −∗ 1^{4} 3_{i} 19∗ 19 2 2 4 22 22  4 22
D_{4}  342  22 4 22 4 22 2 2
45. 2_{−} 11_{i} 3_{i}  0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{i} 19∗ 19 2 4 2 22 22  2 22
D_{4}  6  152  22  4 22 2 22 2 4
46. 2_{−} 11_{i} 3_{i}  0 0 0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{i} 19∗ 19 2 4 2 22  2
D_{4}  6  38  22 22 4 22 2  2 4