Low Degree Fields Ramifying Above {3, 13, 23}

Discriminant class: −299

∞ 3 13 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 23  2 5 7 11 17 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_{i} 0 -9 -180 1^{3} 4_{+} 1  1^{2} 1_{+} 13 13 1^{2} 1_{i} 23∗ 23∗ 2  3 3 2  3 2
S_{3} 3  3 2 2 2 2 2 2 3

Discriminant class: −39

∞ 3 13 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 13 Over p = 23 2 5 7 11 17 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} 1_{i} 1_{+} 2_{+} -1  8 -29 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2 3 2 2 2 2
S_{3}  2 3 3  2 3 3 2
3. 1_{k} 3_{i} 1_{+} 2_{+} 0 -69 -276 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  2
4. 1_{k} 3_{i} 1_{+} 2_{+} 0 69 -115 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 3 2 3 3 2
5. 1_{k} 5_{i} 1_{+} 2_{+} 0 207 -2208 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  2 3 2 2 2 2
S_{3} 3  2 3 3  2 3 3 2
6. 1_{k} 5_{i} 1_{+} 2_{+} 0 -207 -1518 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 2  2 2 2 2
S_{3} 6  2  3 3 2 3 3 2
7. 1_{k} 5_{i} 1_{+} 2_{+} 0 207 -483 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 2 3  2
8. 1_{k} 5_{i} 1_{+} 2_{+} 0 207 -1311 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 3 2 3  2
9. 1_{k} 5_{i} 1_{+} 2_{+} 0 -207 -2967 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2  2 2 2 2
S_{3} 3  2  2  3 2
10. 1_{k} 5_{i} 1_{+} 2_{+} 0 -207 -1173 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2 3 2 2 2 2
S_{3} 6  2 3  3 2  3 2

Discriminant class: −23

∞ 3 13 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 23  2 5 7 11 17 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. 1_{k} 4_{+} 2_{+} 1_{i} 0 -39 -143 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{2} 1_{i} 23∗ 23∗ 3 2 2 2 2 2 3 3
S_{3} 9  2 3 2  2 3 2 2
12. 1_{k} 4_{+} 2_{+} 1_{i} 0 -117 -585 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{2} 1_{i} 23∗ 23∗ 3 2 2 2 2 2  3
S_{3} 9  2 3 2 3 2 3 2 2
13. 1_{k} 4_{+} 2_{+} 1_{i} 0 -39 -442 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{2} 1_{i} 23∗ 23∗  2 2 2 2 2 3
S_{3} 9  2  2 3 2 3 2 2

Discriminant class: −3

∞ 3 13 23 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 13 Over p = 23 2 5 7 11 17 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
14. 1_{k} 1_{i} 2_{+} 2_{+} -1 100 -144 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 3 2 2 2 3
S_{3} 3  3 2 3 2 2 2 3
15. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -299 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 2 3 2 3
S_{3} 6  3 2  2 2 2 3 3
16. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -11661 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 2 3 2 3
S_{3} 3  2  2 2 2 3
17. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -20631 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 3
18. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -2691 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 3 2 2 2 3
S_{3} 6  2 3 2 2 2 3 3
19. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -897 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 2 2 3 2
S_{3} 18  3 2 3 2 2 2 3 3

Discriminant class: 69

∞ 3 13 23 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 13  Over p = 23 2 5 7 11 17 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
20. 0_{+} 5_{i} 2_{+} 1_{k} 0 -234 -429 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{2} 1_{k} 23 23 2 3 2 3 2 2
S_{3} 3  2 2 2 2 3 2 2 2
21. 0_{+} 5_{i} 2_{+} 1_{k} 0 -234 -1365 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{2} 1_{k} 23 23 2 2 3 2 2 3
S_{3} 3  2 2 2 2 3 2 2 2
22. 0_{+} 5_{i} 2_{+} 1_{k} 0 -117 -312 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{2} 1_{k} 23 23 2 3 2 3 3 2 2 3
S_{3} 3  2 2 2 2  2 2 2

Discriminant class: 897

∞ 3 13 23 a_{1} a_{2} a_{3}  Over p = 3  Over p = 13  Over p = 23 2 5 7 11 17 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
23. 0_{+} 1_{i} 1_{+} 1_{k} 0 -897  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 13 13 1^{2} 1_{k} 23 23  2  2  2 2
C_{2} 4  2  2
24. 0_{+} 5_{i} 1_{+} 1_{k} 0 -27 -15 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{2} 1_{k} 23 23 3 2 3 2 3 3 2 2
S_{3}  3  2 3 3  2 3