New Sextic Fields Unramified Outside of {∞, 3, 5}

Discriminant class: −15

∞  3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 1_{k} 9_{i} 7_{+} 0 -15 -5 1^{3} 5_{i} 3∗  1^{5} 7_{+} 5∗  5 32 32 4 42 42 5 6
S_{6}  0 -12 -5 1^{3} 4_{+} ∗  5 32 32 222 5 5 32 33
2. 3_{i} 11_{k} 7_{+} 0 0 -60 1^{6} 11_{k} 3  1^{5} 7_{+} 5∗  5 6 6 4 42 42 5 32
S_{6}  2  45 423 915  5 6 6 2 5 5 6 3

Discriminant class: −3

∞  3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
3. 1_{k} 9_{i} 8_{+} 0 -30 -20 1^{3} 5_{i} 3∗  1^{5} 8_{+} ∗  32 42 6 22 6 5 6 32
S_{6}  225 153 -390 1^{3} 4_{+} 1  33 5 6 42 32 32 32 5
4. 3_{i} 11_{k} 8_{+} 0 0  0 1^{6} 11_{k} 3∗  1^{5} 8_{+} ∗  6 42 32 22 32 5 32 6
S_{6}  4  0 -45 75  3 5 32 42 6 6 6 5

Discriminant class: 1

∞  3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
5. 2_{−}  8_{+} 4_{−} 0 -6 -1 2^{3} 8_{+} 1  1^{4} 3_{−} 5 5 42 42 3 42 42 22 42 22
C_{3}^{2}.C_{4}  9 3 -11  1^{2} 1_{+} 5 5 33 42 33 42 42 42 22 3
6. 2_{−}  8_{+} 4_{−} 0 6 -3 2^{3} 8_{+} 1  1^{4} 3_{−} 5 5 42 42 33 42 42 22 42 22
C_{3}^{2}.C_{4}  9 -9 1  1^{2} 1_{+} 5 5 3 42 3 42 42 42 22 33
7. 2_{−}  6_{−} 8_{+} 0 0 -5 1^{3} 5_{i} 3∗  1^{5} 8_{+} 1  42 42 5 42 42 5 33 5
A_{6}  45 -99 -15 1^{2} 1_{i} 3∗ 3∗  5 5 5 3 3 5 42 5
8. 2_{−}  8_{−} 8_{+} 3 15 25 1^{3} 5_{i} 3∗  1^{5} 8_{+} 1  5 42 3 42 42 42 42 42
A_{6}  0 45 60 1^{3} 3_{i} 3∗  5 42 5 3 5 5 42 42
9. 2_{−} 10_{−} 8_{+} 3 15 25 1^{3} 5_{i} 3∗  1^{5} 8_{+} 1  42 42 5 42 42 5 3 5
A_{6}  3  -75 156 703 1^{3} 5_{i} 3∗  5 5 5 33 33 5 42 5
10. 2_{−} 10_{−} 8_{+} 0 0 75 1^{3} 5_{i} 3∗  1^{5} 8_{+} 1  5 42 33 42 42 42 42 42
A_{6}  0 405 900 1^{3} 5_{i} 3∗  5 42 5 33 5 5 42 42

Discriminant class: 5

∞  3 5 a_{1}  a_{2}  a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 23 29
G  h  a_{4}  a_{5}  a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
11. 2_{−} 8_{+} 5_{−}  0  0  -5 1^{3} 5_{i} 3∗  1^{5} 5_{−} 5  6 6 3 6 4 5 32 42
S_{6}  0 -18  -5 1^{3} 3_{k} 3  5 6 5 222 6 2 42 33
12. 2_{−} 8_{+} 5_{−}  3  0 -10 1^{3} 4_{+} 1  1^{5} 5_{−} 5  6 4 42 32 6 42 32 42
S_{6}  0  18  9 1^{3} 4_{+} ∗  3 32 5 32 6 4 5 5
13. 2_{−} 8_{+} 5_{−} -3  0  10 2^{3} 8_{+} ∗  1^{5} 5_{−} 5  32 4 42 6 32 42 6 42
S_{6}  0  -9  2  33 6 5 6 32 4 5 5

Discriminant class: 5 (continued)

∞  3 5 a_{1}  a_{2}  a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 23 29
G  h  a_{4}  a_{5}  a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
14. 2_{−} 6_{+} 7_{−}  0  0 -10 1^{3} 3_{i} 3∗  1^{5} 7_{−} 5  4 6 5 6 6 5 6 42
S_{6}  0  -3  30 1^{3} 3_{k} 3  42 6 42 6 32 32 3 5
15. 2_{−} 6_{+} 7_{−}  0  0  10 2^{3} 6_{+} ∗  1^{5} 7_{−} 5  4 32 5 32 32 5 32 42
S_{6}  0 -27 -20  42 32 42 32 6 6 33 5
16. 2_{−} 10_{+} 5_{−} -3  0  5 2^{3} 10_{+} ∗  1^{5} 5_{−} 5  32 32 33 32 4 5 6 42
S_{6}  0 -30  35  5 32 5 2 32 222 42 3
17. 2_{−} 8_{+} 7_{−} -3  0  0 1^{3} 5_{i} 3∗  1^{5} 7_{−} 5  32 6 5 6 4 42 32 5
S_{6}  0  3  -4 1^{3} 3_{k} 3  5 6 5 32 32 4 33 42
18. 2_{−} 6_{+} 9_{−}  3  0 -50 1^{3} 3_{k} 3  1^{5} 9_{−} 5  6 4 5 4 6 42 32 22
S_{6}  -75 -15 105 1^{3} 3_{i} 3∗  3 32 5 6 6 6 5 5
19. 2_{−} 6_{+} 9_{−} -2  -5  0 1^{3} 5_{i} 3∗  1^{5} 9_{−} 5  32 6 42 32 6 5 6 5
S_{6}  -30  -3  -9 1^{2} 1_{k} 3 3  33 32 5 32 32 32 5 42
20. 2_{−} 6_{+} 9_{−}  3  0  0 1^{3} 5_{i} 3∗  1^{5} 9_{−} 5  32 6 5 6 32 5 6 5
S_{6}  0 -75 -100 1^{2} 1_{k} 3 3  42 6 42 6 32 32 42 5
21. 2_{−} 6_{+} 9_{−}  0  15  -5 2^{3} 6_{+} ∗  1^{5} 9_{−} 5  6 6 42 6 6 22 6 42
S_{6}  30 -27  5  22 6 42 32 32 32 5 42
22. 2_{−} 6_{+} 9_{−} -1 -20  40 1^{3} 5_{i} 3∗  1^{5} 9_{−} 5  6 6 5 4 222 42 6 3
S_{6}  125 -263 -87 1^{2} 1_{k} 3 3  42 32 5 32 32 32 5 5
23. 2_{−} 6_{+} 9_{−} -3  0 -15 2^{3} 6_{+} ∗  1^{5} 9_{−} 5  32 4 5 4 32 42 6 22
S_{6}  60 -96  88  33 6 5 32 32 32 5 5
24. 2_{−} 6_{+} 9_{−}  0  15 -20 1^{3} 3_{i} 3∗  1^{5} 9_{−} 5  32 32 42 32 32 22 32 42
S_{6}  30  27 -45 1^{3} 3_{k} 3  22 32 42 6 6 6 5 42
25. 2_{−} 10_{+} 7_{−}  0 -15 -30 2^{3} 10_{+} ∗  1^{5} 7_{−} 5  6 32 5 32 4 42 6 5
S_{6}  165  27 -335  5 32 5 6 6 4 3 42
26. 2_{−} 10_{+} 7_{−}  0 -15 -25 1^{6} 10_{+} ∗  1^{5} 7_{−} 5  32 6 42 4 6 5 32 33
S_{6}  0 -15 -50  5 4 5 32 32 32 33 42
27. 2_{−} 10_{+} 7_{−}  0 -15 -55 1^{6} 10_{+} ∗  1^{5} 7_{−} 5  6 32 42 4 32 5 6 3
S_{6}  90 147 -185  5 4 5 6 6 6 3 42
28. 2_{−} 8_{+} 9_{−}  3  0  15 2^{3} 8_{+} ∗  1^{5} 9_{−} 5  4 32 5 32 4 5 6 22
S_{6}  -90  81  18  3 32 5 32 6 6 42 42
29. 2_{−} 8_{+} 9_{−}  0  0 -25 1^{3} 5_{i} 3∗  1^{5} 9_{−} 5  32 32 5 4 6 3 4 5
S_{6}  0  45 -125 1^{3} 3_{k} 3  42 32 5 6 6 32 42 33
30. 2_{−} 8_{+} 9_{−} -3  0 -15 1^{3} 4_{+} ∗  1^{5} 9_{−} 5  4 6 5 6 4 5 32 22
S_{6}  60 -111 158 1^{3} 4_{+} 1  33 6 5 6 32 32 42 42
31. 2_{−} 8_{+} 9_{−}  0 -15 -10 1^{3} 5_{i} 3∗  1^{5} 9_{−} 5  4 32 22 6 32 33 6 3
S_{6}  105  9 -180 1^{3} 3_{k} 3  5 6 42 6 6 6 3 33
32. 2_{−} 10_{+} 9_{−}  0 -15 -90 2^{3} 10_{+} ∗  1^{5} 9_{−} 5  4 6 22 32 6 3 32 33
S_{6}  -195 -189 -80  5 32 42 32 32 32 33 3
33. 2_{−} 10_{+} 9_{−}  0 -15 -15 2^{3} 10_{+} ∗  1^{5} 9_{−} 5  6 6 5 4 32 33 4 5
S_{6}  30 -99 895  42 6 5 32 32 6 42 3
34. 2_{−} 10_{+} 9_{−}  0 -15 -15 2^{3} 10_{+} ∗  1^{5} 9_{−} 5  32 32 5 4 2 42 32 33
S_{6}  30  -9 -80  42 6 5 6 6 6 5 5
35. 2_{−} 10_{+} 9_{−} -3  45 -70 2^{3} 10_{+} ∗  1^{5} 9_{−} 5  6 32 42 6 32 5 32 5
S_{6}  450 -309 -73  3 6 5 6 6 6 5 42
36. 2_{−} 10_{+} 9_{−} -3  0  10 1^{6} 10_{+} ∗  1^{5} 9_{−} 5  32 6 5 4 32 5 6 3
S_{6}  -15 -36 -17  42 32 42 6 4 32 33 5

Discriminant class: 5 (continued)

∞  3 5 a_{1}  a_{2}  a_{3}  Over p = 3  Over p = 5 2 7 11 13 17 19 23 29
G  h  a_{4}  a_{5}  a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
37. 2_{−} 10_{+} 9_{−}  3  0  0 1^{6} 10_{+} ∗  1^{5} 9_{−} 5  6 32 3 6 6 3 4 3
S_{6}  -225 -135 -1155  33 4 5 4 32 32 5 42
38. 2_{−} 10_{+} 9_{−}  0 -15  10 1^{6} 10_{+} ∗  1^{5} 9_{−} 5  4 4 42 6 6 5 32 3
S_{6}  -45 411 -80  5 32 42 4 32 6 22 5
39. 2_{−} 10_{+} 9_{−}  0 -15 -15 2^{3} 10_{+} ∗  1^{5} 9_{−} 5  6 32 5 32 6 5 32 5
S_{6}  30 171 145  42 32 42 32 6 6 42 5
40. 2_{−} 10_{+} 9_{−}  0 -15 -35 1^{6} 10_{+} ∗  1^{5} 9_{−} 5  4 4 42 32 32 5 6 33
S_{6}  -45 -21  -5  5 6 42 4 6 32 22 5
41. 2_{−} 10_{+} 9_{−}  0 -15 -10 1^{6} 10_{+} ∗  1^{5} 9_{−} 5  222 32 5 6 6 5 32 5
S_{6}  2  -45 714 -980  33 6 33 32 32 6 5 3
42. 2_{−} 10_{+} 9_{−}  3  0 -160 1^{6} 10_{+} ∗  1^{5} 9_{−} 5  32 6 42 32 6 42 32 5
S_{6}  210 -369 1168  42 32 33 32 6 4 42 5
43. 2_{−} 10_{+} 9_{−} -3  0 -90 1^{6} 10_{+} ∗  1^{5} 9_{−} 5  32 32 5 32 4 33 6 5
S_{6}  135 -126 753  33 32 3 6 32 4 33 5
44. 2_{−} 10_{+} 9_{−}  0 -15 -10 1^{6} 10_{+} ∗  1^{5} 9_{−} 5  32 6 33 32 32 33 4 33
S_{6}  2  180 -96 -80  3 4 5 4 6 6 5 42
45. 2_{−} 10_{+} 9_{−} -3 -30  50 1^{6} 10_{+} ∗  1^{5} 9_{−} 5  6 32 42 6 32 42 6 5
S_{6}  420 -462 121  42 6 3 6 32 4 42 5
46. 2_{−} 10_{+} 9_{−} -3  0 -15 1^{6} 10_{+} ∗  1^{5} 9_{−} 5  6 6 5 6 4 3 32 5
S_{6}  -540 369 -1257  3 6 33 32 6 4 3 5
47. 2_{−} 10_{+} 9_{−}  0 -15 160 1^{6} 10_{+} ∗  1^{5} 9_{−} 5  6 32 5 4 6 5 32 33
S_{6}  -45 -309 1195  42 6 42 32 4 6 3 5
48. 2_{−} 10_{+} 9_{−}  0 -15  65 1^{6} 10_{+} ∗  1^{5} 9_{−} 5  2 6 5 32 32 5 6 5
S_{6}  180 -1761 -2630  3 32 3 6 6 32 5 33