Low Degree Fields Ramifying Above {3, 5, 17}

Discriminant class: −255

∞ 3 5 17 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 17 2 7 11 13 19 23 29 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_{+} 0 255  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 5 5 1^{2} 1_{+} 17 17  2  2  2
C_{2} 12  2  2 2 2
2. 1_{k} 1_{i} 1_{+} 1_{+} -1 0 -3 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 5 5 1^{2} 1_{+} 17 17 3 3 3 2 3 2 3 2
S_{3}  3 3 2 3 3 2 2 2
3. 1_{k} 5_{i} 1_{+} 1_{+} 0 -27 -99 1^{3} 5_{i} 3∗  1^{2} 1_{+} 5 5 1^{2} 1_{+} 17 17 3 3  2 3 2  2
S_{3} 3  3 3 2 3  2 2 2
4. 1_{k} 5_{i} 1_{+} 1_{+} 0 27 -12 1^{3} 5_{i} 3∗  1^{2} 1_{+} 5 5 1^{2} 1_{+} 17 17  3 3 2 3 2 3 2
S_{3} 3  2 3 3 2 2 2
5. 1_{k} 5_{i} 1_{+} 1_{+} 0 27 -63 1^{3} 5_{i} 3∗  1^{2} 1_{+} 5 5 1^{2} 1_{+} 17 17 3  3 2  2 3 2
S_{3} 6  3 3 2  3 2 2 2

Discriminant class: −3

∞ 3 5 17 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 17 2 7 11 13 19 23 29 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_{i} 2_{+} 2_{+} 0 0 -85 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2  2 3 3 2 2
S_{3} 3  2  2 2 2  3
7. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -425 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 3 2 3  2 2 3
S_{3} 6  3 2 3 2 2 2 3 3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -765 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 3 2 3 3 2 2 3
S_{3} 3  3 2 3 2 2 2  3
9. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1275 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2  2  3 2 2
S_{3} 3  3 2 3 2 2 2 3 3
10. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -255 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 3 2  2 2 3
S_{3} 3  3 2 3 2 2 2  3
11. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -3825 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 3 2 3 3 2 2 3
S_{3} 6  2  2 2 2 3 3

Discriminant class: 5

∞ 3 5 17 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 17 2 7 11 13 19 23 29 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. 0_{+} 4_{+} 1_{+} 2_{+} 0 -102 -391 1^{3} 4_{+} ∗  1^{2} 1_{+} 5 5 1^{3} 2_{+} ∗  2 2 3 2 3 2  3
S_{3}  2  2 2 2 3 3 2

Discriminant class: 17

∞ 3 5 17 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 17 2 7 11 13 19 23 29 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
13. 0_{+} 4_{+} 2_{+} 1_{+} 0 -45 -45 1^{3} 4_{+} ∗  1^{3} 2_{+} ∗  1^{2} 1_{+} 17 17 3 2 2 3 3 2 2 2
S_{3}  2 2 3 3 3  2 3

Discriminant class: 85

∞ 3 5 17 a_{1} a_{2} a_{3} Over p = 3  Over p = 5  Over p = 17  2 7 11 13 19 23 29 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. 0_{+} 4_{+} 1_{−} 1_{−} 0 -12 -1 1^{3} 4_{+} 1  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{−} 17∗ 17∗ 2 3 2 2 3 3 2 2
S_{3} 3  3 2 2 2 2 3 2 2