Low Degree Fields Ramifying Above {3, 7, 17}

Discriminant class: −119

∞ 3 7 17 a_{1} a_{2} a_{3} Over p = 3  Over p = 7  Over p = 17  2 5 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} 4_{+} 1_{k} 1_{−} 0 -3 -19 1^{3} 4_{+} 1  1^{2} 1_{k} 7 7 1^{2} 1_{−} 17∗ 17∗ 3 3 2 2 2 2 2
S_{3} 3 2 3 2 3 2 3

Discriminant class: −7

∞ 3 7 17 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 17 2 5 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
2. 1_{k} 4_{+} 1_{i} 2_{+} 0 51 -68 1^{3} 4_{+} ∗  1^{2} 1_{i} 7∗ 7∗ 1^{3} 2_{+} ∗  2 3 2 2 3 3 2
S_{3} 12  3 2  2 3 2 2 3
3. 1_{k} 4_{+} 1_{i} 2_{+} 0 51 -527 1^{3} 4_{+} ∗  1^{2} 1_{i} 7∗ 7∗ 1^{3} 2_{+} ∗  3 2 3 2 2 3 3 2
S_{3} 3  2 3 2  2 2 3

Discriminant class: −3

∞ 3 7 17 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 17 2 5 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
4. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -119 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 2 3 3 2 2 3
S_{3} 3  2 3 2 2 2
5. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -833 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 2  3 2 2
S_{3} 3  3 2  2 2 2 3 3
6. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -357 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 2 3 3 2 2
S_{3} 6  3 2 3 2 2 2
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -2499 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 2 3 3 2 2 3
S_{3} 6  3 2 3 2 2 2 3 3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -6069 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 2 3 2 2 3
S_{3} 3  2 3 2 2 2 3 3
9. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1071 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 2  2 2 3
S_{3} 6  3 2  2 2 2

Discriminant class: 357

∞ 3 7 17 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 17 2 5 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
10. 0_{+} 1_{i} 1_{k} 1_{+} 0 -357  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{k} 7 7 1^{2} 1_{+} 17 17 2 2  2 2
C_{2} 2  2 2  2
11. 0_{+} 5_{i} 1_{k} 1_{+} 0 -45 -96 1^{3} 5_{i} 3∗  1^{2} 1_{k} 7 7 1^{2} 1_{+} 17 17 2 2 3 2 2 3 3
S_{3}  2 2 3 3 2 3  3