Low Degree Fields Ramifying Above {3, 17, 23}

Discriminant class: −391

∞ 3 17 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 17  Over p = 23 2 5 7 11 13 19 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_{−} 1_{k} 0 -39 -166 1^{3} 4_{+} ∗  1^{2} 1_{−} 17∗ 17∗ 1^{2} 1_{k} 23 23  3 3 3 3 2 2 2
S_{3}  2 2 3 2  2

Discriminant class: −23

∞ 3 17 23 a_{1} a_{2}  a_{3} Over p = 3  Over p = 17  Over p = 23  2 5 7 11 13 19 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_{+} 2_{+} 1_{i} 0 51 -1547 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  1^{2} 1_{i} 23∗ 23∗ 3 2 2 2  2 3 3
S_{3} 3  2  2  2 3 2 2
3. 1_{k} 4_{+} 2_{+} 1_{i} 0 153 -1530 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  1^{2} 1_{i} 23∗ 23∗  2 2 2 3 2  3
S_{3} 18  2 3 2 3 2 3 2 2
4. 1_{k} 4_{+} 2_{+} 1_{i} 0 51 -17 1^{3} 4_{+} 1 1^{3} 2_{+} ∗ 1^{2} 1_{i} 23∗ 23∗ 3 2 2 2 3 2 3
S_{3} 12  2 3 2 3 2 3 2 2

Discriminant class: −3

∞ 3 17 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 17 Over p = 23 2 5 7 11 13 19 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
5. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -391 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2  2  2 3
S_{3} 12  3 2  2 2 2 3 3
6. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -6647 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2 3 2 3 3 2 3
S_{3} 3  3 2 3 2 2 2 3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1173 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 2 3 2 3 3 2 3
S_{3} 12  2 3 2 2 2 3 3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -19941 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2 3 2  3 2
S_{3} 3  3 2  2 2 2 3
9. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -3519 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2 3 2 3 3 2
S_{3} 12  3 2 3 2 2 2 3 3
10. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -26979 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 2 2 3 2 3
S_{3} 6  2 3 2 2 2 3

Discriminant class: 17

∞ 3 17 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 17 Over p = 23 2 5 7 11 13 19 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
11. 0_{+} 4_{+} 1_{+} 2_{+} 0 -207 -1035 1^{3} 4_{+} ∗ 1^{2} 1_{+} 17 17 1^{3} 2_{+} ∗ 3 2 2 2 3 3 2 2
S_{3}  2 2 3 3 3 3 2 3

Discriminant class: 1173

∞ 3 17 23 a_{1}  a_{2} a_{3} Over p = 3  Over p = 17  Over p = 23  2 5 7 11 13 19 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_{+} 1_{k} 1_{+} 1_{i} 0 -1173  1^{2} 1_{k} 3 3 1^{2} 1_{+} 17 17 1^{2} 1_{i} 23∗ 23∗ 2 2  2  2  2
C_{2} 2  2 2  2  2