Low Degree Fields Ramifying Above {2, 3, 5, 7}

Discriminant class: −210

∞ 2 3 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  Over p = 7  11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
1. 1_{k} 3_{i} 1_{i} 1_{−} 1_{i} 0 210  1^{2} 3_{i} −2 −2 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{i} 7∗ 7∗ 2 2 2 2 2
C_{2} 8 2 2 2
2. 1_{k} 3_{i} 3_{i} 1_{−} 1_{i} 0 18 -16 1^{2} 3_{i} −2 −2 1^{3} 3_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{i} 7∗ 7∗ 2 2 2 2 2 3 3 3
S_{3} 5  3 2 3 2 3 3 2 3
3. 1_{k} 3_{i} 5_{i} 1_{−} 1_{i} 0 18 -96 1^{2} 3_{i} −2 −2 1^{3} 5_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{i} 7∗ 7∗ 2 2 2 2 2  3 3
S_{3}  3 2  2  3 2 3
4. 1_{k} 3_{i} 5_{i} 1_{−} 1_{i} 0 -27 -114 1^{2} 3_{i} −2 −2 1^{3} 5_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{i} 7∗ 7∗ 2 2 2 2 2 3  3
S_{3}  2 3 2 3 3 2
5. 1_{k} 3_{i} 5_{i} 1_{−} 1_{i} 0 -27 -306 1^{2} 3_{i} −2 −2 1^{3} 5_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{i} 7∗ 7∗ 2 2 2 2 2 3 3
S_{3} 3 2 3 2 3 2 3

Discriminant class: −105

∞ 2 3 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  Over p = 7  11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
6. 1_{k} 2_{i} 1_{k} 1_{+} 1_{i} 0 105  1^{2} 2_{i} −1 −1 1^{2} 1_{k} 3 3 1^{2} 1_{+} 5 5 1^{2} 1_{i} 7∗ 7∗  2  2 2  2
C_{2} 8  2 2
7. 1_{k} 2_{i} 3_{k} 1_{+} 1_{i} 0 -3 -12 1^{2} 2_{i} −1 −1 1^{3} 3_{k} 3 1^{2} 1_{+} 5 5 1^{2} 1_{i} 7∗ 7∗ 3 3 2 3 2 2 2
S_{3}  3 3 3 3 2 2 3

Discriminant class: −70

∞ 2 3 5 7 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 5 Over p = 7 11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
8. 1_{k} 3_{−} 4_{+} 1_{+} 1_{k} 0 -3 -58 1^{2} 3_{−} 2∗ 2∗ 1^{3} 4_{+} ∗  1^{2} 1_{+} 5 5 1^{2} 1_{k} 7 7 2 2 3  2 2 2 3
S_{3}  2  3 3 3 3 3 3

Discriminant class: −42

∞ 2 3 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  Over p = 7 11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
9. 1_{k} 3_{k} 1_{k} 2_{+} 1_{k} -1 2 -38 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{k} 3 3 1^{3} 2_{+} ∗  1^{2} 1_{k} 7 7 2 3 3 2 3 3 3 2
S_{3}  3 3 2 3 3 3
10. 1_{k} 3_{k} 3_{k} 2_{+} 1_{k} 0 15 -30 1^{2} 3_{k} −2∗ −2∗ 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{k} 7 7 2 3  2  3 3 2
S_{3} 3  3  2 3  3 3 3
11. 1_{k} 3_{k} 3_{k} 2_{+} 1_{k} 0 15 -110 1^{2} 3_{k} −2∗ −2∗ 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{k} 7 7 2  3 2 3  2
S_{3} 3  3 3 2  3 3 3 3

Discriminant class: −35

∞ 2 3 5 7 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 5  Over p = 7 11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
12. 1_{k} 2_{+} 4_{+} 1_{−} 1_{k} 0 -12 -44 1^{3} 2_{+} ∗  1^{3} 4_{+} 1  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{k} 7 7  3 3 2 2 3 2 2
S_{3} 3  2 2  2 2 2 2 3
13. 1_{k} 2_{+} 4_{+} 1_{−} 1_{k} 0 -12 -26 1^{3} 2_{+} ∗  1^{3} 4_{+} 1  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{k} 7 7 3  2 2 3 2 2
S_{3} 3  2 2 3 2 2 2 2 3

Discriminant class: −10

∞ 2 3 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  Over p = 7 11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
14. 1_{k} 3_{k} 4_{+} 1_{−} 2_{+} 0 42 -112 1^{2} 3_{k} −2∗ −2∗ 1^{3} 4_{+} ∗  1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} 1  3 3 2 3  2 2 3
S_{3} 3  2 3  3 2 2 2
15. 1_{k} 3_{k} 4_{+} 1_{−} 2_{+} 0 63 -126 1^{2} 3_{k} −2∗ −2∗ 1^{3} 4_{+} ∗  1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} 1  3  2  3 2 2 3
S_{3} 6  3 2  3 3 2 2 2

Discriminant class: −5

∞ 2 3 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  Over p = 7 11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
16. 1_{k} 2_{i} 4_{+} 1_{+} 2_{+} 0 -63 -252 1^{2} 2_{i} −∗ −∗ 1^{3} 4_{+} 1  1^{2} 1_{+} 5 5 1^{3} 2_{+} 1  2 2 2 2  3 2 2
S_{3} 9  3 3 3 2 2  2
17. 1_{k} 2_{i} 4_{+} 1_{+} 2_{+} 0 42 -28 1^{2} 2_{i} −∗ −∗ 1^{3} 4_{+} 1  1^{2} 1_{+} 5 5 1^{3} 2_{+} 1  2 2 2 2 3  2 2
S_{3} 9  3 3 2 2 3 3 2

Discriminant class: −3

∞ 2 3 5 7 a_{1} a_{2}  a_{3} Over p = 2  Over p = 3  Over p = 5  Over p = 7 11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
18. 1_{k} 2_{+} 1_{i} 2_{+} 2_{+} -1 -23  57 1^{3} 2_{+} ∗  1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2  2  2 2 3 3
S_{3} 3  2 3 2 2 2 3  2
19. 1_{k} 2_{+} 3_{i} 2_{+} 2_{+} 0 0 -140 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2  2 2
S_{3} 9  2  2 2 2 3 3 2
20. 1_{k} 2_{+} 3_{i} 2_{+} 2_{+} 0 0 -490 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2 3 2 2 3 3
S_{3} 9  2 3 2 2 2 3 3 2
21. 1_{k} 2_{+} 3_{i} 2_{+} 2_{+} 0 0 -70 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2  2 3 2 2  3
S_{3} 9  2  2 2 2  2
22. 1_{k} 2_{+} 5_{i} 2_{+} 2_{+} 0 0 -1050 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2 3 2 2
S_{3} 18  2 3 2 2 2 3  2
23. 1_{k} 2_{+} 5_{i} 2_{+} 2_{+} 0 0 -2100 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2 3 2 2  3
S_{3} 9  2 3 2 2 2 3 3 2
24. 1_{k} 2_{+} 5_{i} 2_{+} 2_{+} 0 0 -630 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2 3 2 2 3
S_{3} 9  2 3 2 2 2  2
25. 1_{k} 2_{+} 5_{i} 2_{+} 2_{+} 0 0 -1470 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2  2  2 2 3
S_{3} 9  2  2 2 2 3 3 2

Discriminant class: −3 (continued)

∞ 2 3 5 7 a_{1} a_{2}  a_{3} Over p = 2  Over p = 3  Over p = 5  Over p = 7 11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
26. 1_{k} 2_{+} 5_{i} 2_{+} 2_{+} 0 0 -2940 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2 3 2 2 3 3
S_{3} 18  2  2 2 2 3  2
27. 1_{k} 2_{+} 5_{i} 2_{+} 2_{+} 0 0 -1260 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2  2 3 2 2 3 3
S_{3} 9  2 3 2 2 2 3 3 2
28. 1_{k} 2_{+} 5_{i} 2_{+} 2_{+} 0 0 -420 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 3 2 3 2 2 3 3
S_{3} 9  2 3 2 2 2 3 3 2
29. 1_{k} 2_{+} 5_{i} 2_{+} 2_{+} 0 0 -210 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2  2 2 3 3
S_{3} 18  2 3 2 2 2  2

Discriminant class: 2

∞ 2 3 5 7 a_{1} a_{2}  a_{3} Over p = 2  Over p = 3  Over p = 5  Over p = 7 11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
30. 0_{+} 3_{+} 4_{+} 2_{+} 2_{+} 0 -210 -1120 1^{2} 3_{+} 2 2 1^{3} 4_{+} ∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 3 2 3 2 3 2
S_{3} 3  2  2 2 2 2 3

Discriminant class: 7

∞ 2 3 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  Over p = 7 11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
31. 0_{+} 2_{i} 4_{+} 2_{+} 1_{k} 0 -75 -100 1^{2} 2_{i} −1 −1 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  1^{2} 1_{k} 7 7 2 2 2 3 2  3 3
S_{3} 3  2 2  3 3 2 2 2

Discriminant class: 15

∞ 2 3 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  Over p = 7 11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
32. 0_{+} 2_{i} 3_{i} 1_{−} 2_{+} 0 -42 -84 1^{2} 2_{i} −1 −1 1^{3} 3_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} 1  3 2 3 2 2 2 2 2
S_{3} 3  2  2 3 3  3 3
33. 0_{+} 2_{i} 5_{i} 1_{−} 2_{+} 0 -63 -42 1^{2} 2_{i} −1 −1 1^{3} 5_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} 1  3 2  2 2 2 2 2
S_{3} 3  2 3 2 3 3 3 3
34. 0_{+} 2_{i} 5_{i} 1_{−} 2_{+} 0 -63 -168 1^{2} 2_{i} −1 −1 1^{3} 5_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} 1  2 3 2 2 2 2 2
S_{3} 3  2 3 2  3 3  3

Discriminant class: 30

∞ 2 3 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  Over p = 7 11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
35. 0_{+} 3_{i} 3_{k} 1_{+} 2_{+} 0 -42 -56 1^{2} 3_{i} −2 −2 1^{3} 3_{k} 3  1^{2} 1_{+} 5 5 1^{3} 2_{+} 1  2 3 3 3 2  2
S_{3} 3  2 2 2 2 2 2 2 3

Discriminant class: 42

∞ 2 3 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  Over p = 7  11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
36. 0_{+} 3_{−} 3_{i} 2_{+} 1_{i} 0 -45 -110 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{i} 3∗ 1^{3} 2_{+} ∗ 1^{2} 1_{i} 7∗ 7∗ 3 3 3 2 3 2 2
S_{3}  2 3 3 2 3 2 2
37. 0_{+} 3_{−} 5_{i} 2_{+} 1_{i} 0 -90 -240 1^{2} 3_{−} 2∗ 2∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{2} 1_{i} 7∗ 7∗ 3 3  3 2 3 2 2
S_{3}  3 2  2 3 2 2
38. 0_{+} 3_{−} 5_{i} 2_{+} 1_{i} 0 -45 -30 1^{2} 3_{−} 2∗ 2∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{2} 1_{i} 7∗ 7∗ 3  3 3 2  2 2
S_{3}  3 2 3 3 2  2 2

Discriminant class: 210

∞ 2 3 5 7 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 5  Over p = 7 11 13 17 19 23 29 31 37
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 41 43 47 53 59 61 67 71
39. 0_{+} 3_{+} 1_{k} 1_{−} 1_{k} 0 -210  1^{2} 3_{+} 2 2 1^{2} 1_{k} 3 3 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{k} 7 7  2 2  2
C_{2} 4  2 2 2  2