Quartic Fields Ramifying Above {2, 13}

Discriminant class: −26

∞  2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  3 5 7 11 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_{+} 13∗ 13∗ 22 22 22 2 22 2 2 4
D_{4}  2  -104  22 22 4 22 22 4 2 4
2. 1_{k} 11_{i} 3_{+}  0 0  0 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{+} 13∗ 13∗ 22  22 2 22 2 2 4
D_{4}  2  -26  22  4 22 22 4 2 4
3. 1_{k} 9_{i} 1_{+}  0 0 -4 1^{4} 9_{i} −2∗ −∗ 1^{2} 1_{+} 13 13 3 3 3 4 3 2 4 4
S_{4} 1 2 0_{+} ∗ ∗ 22 3 4 3 3 4 4 4
4. 1_{k} 11_{k} 1_{−}  0 -8 -8 1^{4} 11_{k} −2∗ −2 1^{2} 1_{−} 13∗ 13∗ 3 3 3 4 3 4 2 4
S_{4} 2 22 3 4 3 3 2 2 4
5. 1_{k} 11_{i} 1_{+}  0 -4 -8 1^{4} 11_{i} −2∗ 2∗ 1^{2} 1_{+} 13 13 3 3 3 2 3 4 4 2
S_{4}  2  -6  2 0_{+} ∗  ∗ 22 3 2 3 3 4 4 2
6. 1_{k} 5_{i} 3_{+}  2 8 20 1^{2} 3_{+} 2  2 1^{4} 3_{+} 13∗ 13∗ 3 3 3 4 3 2 4 4
S_{4}  2  -4  1^{2} 2_{i} −∗ −∗  22 3 2 3 3 4 2 2
7. 1_{k} 9_{i} 3_{+}  0 0 -52 1^{4} 9_{i} −2∗ −∗ 1^{4} 3_{+} 13∗ 13∗ 3 3 3 2 3 2 2 2
S_{4} 4 -143 22 3 4 3 3 2 4 4
8. 1_{k} 11_{i} 3_{+}  0 0 104 1^{4} 11_{i} −2∗ −2 1^{4} 3_{+} 13∗ 13∗ 3 3 3 2 3 4 4 2
S_{4} 4 -78 22 3 4 3 3 4 2 4
9. 1_{k} 11_{i} 3_{+}  0 -52 -104 1^{4} 11_{i} −2∗ 2∗ 1^{4} 3_{+} 13∗ 13∗ 3 3 3 4 3 4 2 4
S_{4} 4 234 3 2 3 3 2 4 2

Discriminant class: −13

∞ 2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13 3 5 7 11 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} 3_{−} 0 0 0 2^{2} 4_{k} −∗ ∗ 1^{4} 3_{−} 13 13 2 4 22 22  22 2 22
D_{4} -13 22 4 4 2 22 22
11. 1_{k} 6_{k} 3_{−} 0 0 0 2^{2} 6_{k} −∗ ∗ 1^{4} 3_{−} 13 13 2 4 22 22  22 2
D_{4} -52 22 4 4 2 22 22 22 22

Discriminant class: −2

∞  2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13 3 5 7 11 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
12. 1_{k} 11_{i} 2_{+}  0 0 0 1^{4} 11_{i} −2 2 2^{2} 2_{+} ∗ ∗ 22 4 2 22 22 22 2 4
D_{4} 2 -338 2 4 22 2 4 22 4

Discriminant class: −1

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

Discriminant class: −1 (continued)

∞  2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  3 5 7 11 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} 6_{i} 2_{+}  0 6 0 2^{2} 6_{i} −1 ∗ 1^{2} 1_{+} 13 13 2 22 4 4 22 4 2
D_{4}  -4  1^{2} 1_{+} 13 13 4 22 22 2 4 22 4 22
15. 1_{k} 10_{k} 2_{−}  0 26 0 1^{4} 10_{k} −1 2 2^{2} 2_{−} 1  V 4 22 2 4 22 4 2 22
D_{4}  2  -169  2 22 22 4 2 22 4 22
16. 1_{k} 10_{i} 2_{+}  0 -2 0 1^{4} 10_{i} −1 2∗ 1^{2} 1_{−} 13∗ 13∗ 4 22 4 2  2 2 22
D_{4}  2  -25  1^{2} 1_{−} 13∗ 13∗ 4  22 4 4 22 2 22
17. 1_{k} 10_{k} 2_{−}  0 10 0 1^{4} 10_{k} −1 2∗ 2^{2} 2_{−} 1  V 4  4 2  2 2 22
D_{4}  2  -1  4 22 22 4 4 22 2 22
18. 1_{k} 6_{i} 2_{+}  0 -2 -8 2^{2} 6_{i} −1 ∗ 1^{3} 2_{+} 1  4 3 2 2 3 4 4 3
S_{4}  4  4 3 3 2 4 3 2 3
19. 1_{k} 10_{i} 2_{+}  0 -2 -8 1^{4} 10_{i} −1 2 1^{3} 2_{+} 1  2 3 4 4 3 2 4 3
S_{4}  2  -9  4 3 3 2 2 3 4 3
20. 1_{k} 10_{i} 2_{+}  0 -6 -8 1^{4} 10_{i} −1 2∗ 1^{3} 2_{+} 1  4 3 4 4 3 4 2 3
S_{4}  2  23  2 3 3 2 4 3 4 3

Discriminant class: 1

∞ 2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  3 5 7 11 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
21. 2_{−} 4_{−} 2_{+} 0 7 0 2^{2} 4_{−} 1 {−1 \atop −∗} 1^{2} 1_{+} 13 13 22 22 22 22  22 22
V_{4}  9  1^{2} 1_{+} 13 13 22 22 22 22 22  22
22. 0_{+} 6_{−} 2_{−} -2 -9 10 2^{2} 6_{−} 1  {∗ \atop 2} 2^{2} 2_{−} 1  V 22 22 22 22  22  22
V_{4}  -1  22 22 22 22 22 22 22 22
23. 2_{−} 6_{+} 2_{−} 2 -1 -2 2^{2} 6_{+} 1 {−2 \atop −2∗} 2^{2} 2_{−} 1  V  22 22 22  22 22 22
V_{4} 3  27  22 22 22  22 22 22 22
24. 2_{−} 8_{+} 2_{−} 0 -12 0 1^{4} 8_{+} 1 {−2 \atop −∗} 2^{2} 2_{−} 1  V 22 22 22  22 22
V_{4} 2  49  22 22 22 22 22 22  22
25. 2_{−} 8_{−} 2_{+} 0 0 0 1^{4} 8_{−} 1 {−2∗ \atop −1} 1^{2} 1_{−} 13∗ 13∗ 22  22 22  22 22 22
V_{4} 6  169  1^{2} 1_{−} 13∗ 13∗ 22  22 22 22 22 22 22
26. 2_{−} 8_{+} 2_{−} 0 12 0 1^{4} 8_{+} 1 {−2∗ \atop −∗} 2^{2} 2_{−} 1  V 22 22  22  22 22 22
V_{4} 6  49  22 22 22  22 22 22
27. 2_{−} 6_{−} 2_{+} 2 2 6 1^{4} 6_{−} 1  1^{3} 2_{+} 1  3 22 3 3 3 3 3 3
A_{4}  7  22 3 3 3  22 3 3

Discriminant class: 2

∞  2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  3 5 7 11 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
28. 2_{−} 11_{−} 2_{+} 0 52 0 1^{4} 11_{−} 2 2 2^{2} 2_{+} ∗  ∗ 4 4 22 4  4  4
C_{4} 10  338  4  4  4 4 4
29. 0_{+} 11_{+} 2_{+} 0 -52 0 1^{4} 11_{+} 2 2 2^{2} 2_{+} ∗  ∗ 4 4  4  4 22 4
C_{4}  2  338  22 4  4 22 4 4 4
30. 2_{−} 9_{+} 2_{−} 0 26 0 1^{4} 9_{+} 2 −1 1^{2} 1_{−} 13∗ 13∗ 4 2 22 4 22 4 22 2
D_{4}  2  338  1^{2} 1_{+} 13 13 22 2 22 4 22 2 4 2

Discriminant class: 2 (continued)

∞  2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  3 5 7 11 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_{−} 2_{+} 0 0 0 1^{4} 11_{−} 2 −2 2^{2} 2_{+} ∗  ∗ 2 4 22 2 22 2 22 4
D_{4}  2  338  22 4  2 22 4 2 4

Discriminant class: 13

∞ 2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  3 5 7 11 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. 0_{+} 4_{−} 3_{−} 0 -13 0 2^{2} 4_{−} ∗ ∗ 1^{4} 3_{−} 13 13 22 4 4 4 22 4
C_{4} 13 4 4 4 4 4
33. 2_{−} 6_{+} 3_{−} 0 26 0 2^{2} 6_{+} ∗ ∗ 1^{4} 3_{−} 13 13 22 4 4 4 22 4 22 22
C_{4} 2  52  4 4 4  4 22 4 22
34. 0_{+} 6_{−} 3_{−} 0 -26 0 2^{2} 6_{−} ∗ ∗ 1^{4} 3_{−} 13 13  4 4 4 22 4  22
C_{4}  52  4 4 4 22 4 22 4 22
35. 2_{−} 6_{+} 1_{−} 2 4 6 1^{4} 6_{+} ∗ −1 1^{2} 1_{−} 13∗ 13∗ 22 2 4 4 22 4 22 22
D_{4} 5 2 0_{+} ∗ ∗ 4 2 2 22 4 4
36. 2_{−} 8_{−} 1_{+} 0 -4 0 1^{4} 8_{−} ∗ −1 1^{2} 1_{+} 13 13 22 2 4 4 22 4 22
D_{4} 13 4 2 2 22 4 22 4 22
37. 2_{−} 6_{+} 3_{−} 2 8 20 1^{4} 6_{+} ∗ −∗ 1^{4} 3_{−} 13 13 22 4 2 2  2 22 22
D_{4} 4 22 2 4 4 22 2 2
38. 2_{−} 8_{+} 3_{−} 0 0 0 1^{4} 8_{+} ∗ −∗ 1^{4} 3_{−} 13 13 22 4 2 2  2 22
D_{4} 4  13  2 4 4 22 2 22 2 22

Discriminant class: 26

∞  2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  3 5 7 11 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. 2_{−} 11_{−} 3_{+} 0 52 0 1^{4} 11_{−} 2∗ 2∗ 1^{4} 3_{+} 13∗ 13∗ 4 22 4  22  4
C_{4} 20  650  4  4 4 4 4  4
40. 2_{−} 11_{−} 3_{+} 0 52 0 1^{4} 11_{−} 2∗ 2∗ 1^{4} 3_{+} 13∗ 13∗ 4  4 22 22 22  4
C_{4} 20  26  4 22 4 4 4 4 22 4
41. 0_{+} 11_{+} 3_{+} 0 -52 0 1^{4} 11_{+} 2∗ 2∗ 1^{4} 3_{+} 13∗ 13∗ 4 22 4 22 22 22 22 4
C_{4}  2  650  4  4 4 4 4 22 4
42. 0_{+} 11_{+} 3_{+} 0 -52 0 1^{4} 11_{+} 2∗ 2∗ 1^{4} 3_{+} 13∗ 13∗ 4  4  22  22 4
C_{4}  2  26  4 22 4 4 4 4  4
43. 2_{−} 9_{−} 1_{+} 0 -2 0 1^{4} 9_{−} 2∗ −1 1^{2} 1_{+} 13 13 4  4 22  22 22 2
D_{4}  26  2 0_{+} ∗  ∗ 4 22 2 4 4 2 22 2
44. 2_{−} 9_{+} 1_{−} 0 -10 0 1^{4} 9_{+} 2∗ −1 1^{2} 1_{−} 13∗ 13∗ 4 22 4 22  22 22 2
D_{4} 26 4 2 4 4 2 22 2
45. 2_{−} 11_{−} 3_{+} 0 0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{+} 13∗ 13∗ 2 22 2 22 22 22 22 4
D_{4} 12 26 2 22 4 2 2 4 22 4
46. 2_{−} 11_{−} 3_{+} 0 0 0 1^{4} 11_{−} 2∗ −2∗ 1^{4} 3_{+} 13∗ 13∗ 2  2 22 22 22 22 4
D_{4} 12  104  2  4 2 2 4 22 4