Low Degree Fields Ramifying Above {3, 13, 17}

Discriminant class: −663

∞ 3 13 17 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 17  2 5 7 11 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_{k} 1_{+} 1_{−} 0 663  1^{2} 1_{k} 3 3 1^{2} 1_{+} 13 13 1^{2} 1_{−} 17∗ 17∗  2  2 2
C_{2} 16  2  2  2 2
2. 1_{k} 3_{k} 1_{+} 1_{−} 0 -3 -15 1^{3} 3_{k} 3  1^{2} 1_{+} 13 13 1^{2} 1_{−} 17∗ 17∗ 3 2 3 2 2 3 3 3
S_{3} 2 3 2 3 2 2

Discriminant class: −51

∞ 3 13 17 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 17  2 5 7 11 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
3. 1_{k} 1_{k} 2_{+} 1_{−} -1 9 12 1^{2} 1_{k} 3 3 1^{3} 2_{+} 1  1^{2} 1_{−} 17∗ 17∗ 2 3 2 3  3 3 2
S_{3} 9  2  3 2 2 2 2 3
4. 1_{k} 3_{k} 2_{+} 1_{−} 0 78 -39 1^{3} 3_{k} 3  1^{3} 2_{+} 1  1^{2} 1_{−} 17∗ 17∗ 2  2 3 3 3  2
S_{3} 15  2 3  2 2 2 2 3
5. 1_{k} 3_{k} 2_{+} 1_{−} 0 39 -52 1^{3} 3_{k} 3  1^{3} 2_{+} 1  1^{2} 1_{−} 17∗ 17∗ 2 3 2 3 3  3 2
S_{3} 3  2 3 3 2 2 2 2

Discriminant class: −39

∞ 3 13 17 a_{1} a_{2} a_{3} Over p = 3 Over p = 13 Over p = 17 2 5 7 11 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} 1_{i} 1_{+} 2_{+} -1  6 -63 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 2
7. 1_{k} 3_{i} 1_{+} 2_{+} 0 51 -119 1^{3} 3_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2 3 2 2 2 2
S_{3} 3  2 3  3 2 3  2
8. 1_{k} 3_{i} 1_{+} 2_{+} 0 51 -340 1^{3} 3_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  2 3 2 2 2 2
S_{3} 3  2 3 3 3 2 3 3 2
9. 1_{k} 5_{i} 1_{+} 2_{+} 0 153 -102 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 2 3 2 2 2 2
S_{3} 3  2  2 3 3 2
10. 1_{k} 5_{i} 1_{+} 2_{+} 0 -153 -816 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 3 2  2
11. 1_{k} 5_{i} 1_{+} 2_{+} 0 153 -1887 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3 3 2  2 2 2 2
S_{3} 3  2 3  3 2 3 3 2
12. 1_{k} 5_{i} 1_{+} 2_{+} 0 153 -561 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 3 2
13. 1_{k} 5_{i} 1_{+} 2_{+} 0 -153 -1173 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  3  2 3 2 2 2 2
S_{3} 3  2  3 3 2 3  2
14. 1_{k} 5_{i} 1_{+} 2_{+} 0 -153 -1479 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 2  3 2

Discriminant class: −3

∞ 3 13 17 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 13 Over p = 17 2 5 7 11 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
15. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -221 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2  2 2
S_{3} 3  2 3 2 2 2  3
16. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -2873 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 2 2 3
S_{3} 6  3 2  2 2 2 3
17. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -8619 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2  2 2
S_{3} 6  3 2 3 2 2 2 3
18. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -11271 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 2 3 2 2 3
S_{3} 6  2 3 2 2 2 3
19. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1989 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 3 2 2 3
S_{3} 3  3 2  2 2 2  3
20. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -663 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 2 2 3
S_{3} 3  3 2 3 2 2 2  3

Discriminant class: 17

∞ 3 13 17 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 17 2 5 7 11 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
21. 0_{+} 4_{+} 2_{+} 1_{+} 0 -39 -13 1^{3} 4_{+} ∗  1^{3} 2_{+} 1  1^{2} 1_{+} 17 17 3 2 2 2  2 2 2
S_{3} 3  2 2  3 3 3 2