Low Degree Fields Ramifying Above {3, 5, 29}

Discriminant class: −435

∞ 3 5 29 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5  Over p = 29  2 7 11 13 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_{i} 1_{−} 1_{−} 0 435  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{−} 29∗ 29∗ 2 2  2 2 2  2
C_{2} 4  2  2 2 2
2. 1_{k} 3_{i} 1_{−} 1_{−} 0 3 -24 1^{3} 3_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{−} 29∗ 29∗ 2 2 3 2 2 2 3 2
S_{3} 3 2 3 2 2 2
3. 1_{k} 5_{i} 1_{−} 1_{−} 0 -18 -183 1^{3} 5_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{−} 29∗ 29∗ 2 2 3 2 2 2 3 2
S_{3}  3 3 2 3 2 2 2
4. 1_{k} 5_{i} 1_{−} 1_{−} 0 27 -48 1^{3} 5_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{−} 29∗ 29∗ 2 2  2 2 2 3 2
S_{3}  3 3 3 2  2 2 2
5. 1_{k} 5_{i} 1_{−} 1_{−} 0 18 -21 1^{3} 5_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{−} 29∗ 29∗ 2 2 3 2 2 2  2
S_{3}  3 3 3 2 3 2 2 2

Discriminant class: −87

∞ 3 5 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 5  Over p = 29  2 7 11 13 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} 3_{k} 2_{+} 1_{−} 0 -15 -35 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{−} 29∗ 29∗ 3  3  2 2 2
S_{3} 3  2 3 2 3 2 2 2
7. 1_{k} 3_{k} 2_{+} 1_{−} 0 -15 -110 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{−} 29∗ 29∗  3  3 3 2 2 2
S_{3} 3  2 3 2  2 2 2 3
8. 1_{k} 3_{k} 2_{+} 1_{−} 0 15 -15 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{−} 29∗ 29∗ 3 3 3 3 3 2 2 2
S_{3} 3  2 3 2 3 2 2 2 3

Discriminant class: −15

∞ 3 5 29 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 29 2 7 11 13 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
9. 1_{k} 1_{k} 1_{−} 2_{+} -1 10 15 1^{2} 1_{k} 3 3 1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} ∗  3 2 2 2  3 3 3
S_{3}  2 2 2 3  2 3 2
10. 1_{k} 3_{k} 1_{−} 2_{+} 0 87 -232 1^{3} 3_{k} 3  1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} ∗  2 2 2 3 3  3
S_{3} 18  2 2 2 3 3 2  2
11. 1_{k} 3_{k} 1_{−} 2_{+} 0 87 -87 1^{3} 3_{k} 3  1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} ∗  3 2 2 2 3  3
S_{3} 9  2 2 2 3 3 2 3 2

Discriminant class: −3

∞ 3 5 29 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 29 2 7 11 13 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
12. 1_{k} 1_{i} 2_{+} 2_{+} 0 0 -145 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 3 2 3 2  2 3
S_{3}  3 2  2 2 2  3
13. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -725 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 3 2 3 2 3 2 3
S_{3} 3  3 2 3 2 2 2 3 3

Discriminant class: −3 (continued)

∞ 3 5 29 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 5  Over p = 29 2 7 11 13 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
14. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -2175 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 3 2 3 2 3 2 3
S_{3} 3  2 3 2 2 2 3 3
15. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -6525 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2  2  2  2
S_{3} 3  3 2  2 2 2 3 3
16. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1305 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 3 2  2 3 2 3
S_{3} 12  2 3 2 2 2  3
17. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -435 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2  2 3 2 3 2
S_{3} 3  3 2 3 2 2 2  3