Low Degree Fields Ramifying Above {3, 13, 29}

Discriminant class: −1131

∞ 3 13 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 29  2 5 7 11 17 19 23 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d  s 37 41 43 47 53 59 61 67
1. 1_{k} 1_{k} 1_{+} 1_{−} 0 1131  1^{2} 1_{k} 3 3 1^{2} 1_{+} 13 13 1^{2} 1_{−} 29∗ 29∗ 2  2 2  2
C_{2} 8  2 2 2 2  2 2
2. 1_{k} 3_{k} 1_{+} 1_{−} 0 12 -11 1^{3} 3_{k} 3  1^{2} 1_{+} 13 13 1^{2} 1_{−} 29∗ 29∗ 2 3 2 2 3 3 2 3
S_{3} 2 2 2 2 2 2

Discriminant class: −87

∞ 3 13 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 29  2 5 7 11 17 19 23 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d  s 37 41 43 47 53 59 61 67
3. 1_{k} 1_{k} 2_{+} 1_{−} -1 22 51 1^{2} 1_{k} 3 3 1^{3} 2_{+} 1  1^{2} 1_{−} 29∗ 29∗ 3 2 3  2 2 2
S_{3} 3  2 3 2  2 2 2 3
4. 1_{k} 1_{k} 2_{+} 1_{−} -1 -4 25 1^{2} 1_{k} 3 3 1^{3} 2_{+} 1  1^{2} 1_{−} 29∗ 29∗ 3 2  3 3 2 2 2
S_{3} 6  2 3 2 3 2 2 2 3
5. 1_{k} 1_{k} 2_{+} 1_{−} -1 -4 -92 1^{2} 1_{k} 3 3 1^{3} 2_{+} 1 1^{2} 1_{−} 29∗ 29∗ 2 3 3 3 2 2 2
S_{3} 3  2 3 2 3 2 2 2

Discriminant class: −39

∞ 3 13 29 a_{1} a_{2} a_{3} Over p = 3 Over p = 13 Over p = 29 2 5 7 11 17 19 23 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
6. 1_{k} 1_{i} 1_{+} 2_{+} -1 10 -72 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 2 3 2 2 2 2
S_{3}  2 3  3 2 3 3 2
7. 1_{k} 3_{i} 1_{+} 2_{+} 0 -87 -609 1^{3} 3_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3  2 3 2 2 2 2
S_{3} 3  2 3 3  2 3  2
8. 1_{k} 3_{i} 1_{+} 2_{+} 0 87 -29 1^{3} 3_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2 3 2 2 2 2
S_{3} 6  2 3 3 3 2  3 2
9. 1_{k} 5_{i} 1_{+} 2_{+} 0 261 -1479 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2 3 2 2 2 2
S_{3} 3  2  3 3 2  3 2
10. 1_{k} 5_{i} 1_{+} 2_{+} 0 261 -1914 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 2  2 2 2 2
S_{3} 15  2 3 3  2  2
11. 1_{k} 5_{i} 1_{+} 2_{+} 0 -261 -1653 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2  2 2 2 2
S_{3} 6  2 3  3 2 3 3 2
12. 1_{k} 5_{i} 1_{+} 2_{+} 0 261 -3045 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3  2  2 2 2 2
S_{3} 3  2 3 3 3 2 3 3 2
13. 1_{k} 5_{i} 1_{+} 2_{+} 0 261 -3741 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2 3 2 2 2 2
S_{3} 3  2  2 3  2
14. 1_{k} 5_{i} 1_{+} 2_{+} 0 -261 -1740 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  2 3 2 2 2 2
S_{3} 21  2  3 3 2 3 3 2

Discriminant class: −3

∞ 3 13 29 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 13 Over p = 29 2 5 7 11 17 19 23 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
15. 1_{k} 1_{i} 2_{+} 2_{+} 0 0 -377 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 2 2 3 2 3
S_{3} 3  3 2 3 2 2 2
16. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -4901 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 2  2 3
S_{3} 24  3 2  2 2 2 3 3
17. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -14703 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 2 3 2 3
S_{3} 18  2 3 2 2 2 3 3
18. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -32799 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 3 2 2 3 2
S_{3} 15  3 2 3 2 2 2 3 3
19. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -3393 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 2  2 3
S_{3} 3  2  2 2 2
20. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1131 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 2 3 2
S_{3} 12  3 2 3 2 2 2