Low Degree Fields Ramifying Above {2, 3, 17}

Discriminant class: −102

∞ 2 3 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 17  5 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} 3_{−} 1_{i} 1_{−} 0 102  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{−} 17∗ 17∗ 2 2 2 2 2  2 2
C_{2} 4  2 2  2
2. 1_{k} 3_{−} 3_{i} 1_{−} 0 -6 -24 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{i} 3∗  1^{2} 1_{−} 17∗ 17∗ 2 2 2 2 2 3 2 2
S_{3}  3 3 2 2 3 3  2
3. 1_{k} 3_{−} 5_{i} 1_{−} 0 45 -78 1^{2} 3_{−} 2∗ 2∗ 1^{3} 5_{i} 3∗  1^{2} 1_{−} 17∗ 17∗ 2 2 2 2 2  2 2
S_{3}  3 3 2 2 3 3 3 2
4. 1_{k} 3_{−} 5_{i} 1_{−} 0 -27 -150 1^{2} 3_{−} 2∗ 2∗ 1^{3} 5_{i} 3∗  1^{2} 1_{−} 17∗ 17∗ 2 2 2 2 2 3 2 2
S_{3} 3 2 2 3 3 2
5. 1_{k} 3_{−} 5_{i} 1_{−} 0 27 -90 1^{2} 3_{−} 2∗ 2∗ 1^{3} 5_{i} 3∗  1^{2} 1_{−} 17∗ 17∗ 2 2 2 2 2 3 2 2
S_{3} 3 2 2 3 3 2

Discriminant class: −51

∞ 2 3 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 17  5 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} 2_{+} 1_{k} 1_{−} -1 1 -3 1^{3} 2_{+} ∗  1^{2} 1_{k} 3 3 1^{2} 1_{−} 17∗ 17∗ 3 2 3 3 3 3  2
S_{3}  2 3 3 2 2 2 2
7. 1_{k} 2_{+} 3_{k} 1_{−} 0 6 -6 1^{3} 2_{+} ∗  1^{3} 3_{k} 3  1^{2} 1_{−} 17∗ 17∗ 3 2 3  3  3 2
S_{3} 3  2  2 2 2 2 3
8. 1_{k} 2_{+} 3_{k} 1_{−} 0 -6 -10 1^{3} 2_{+} ∗  1^{3} 3_{k} 3  1^{2} 1_{−} 17∗ 17∗  2 3 3  3 3 2
S_{3} 3  2 3 3 2 2 2 2 3

Discriminant class: −34

∞ 2 3 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 17 5 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
9. 1_{k} 3_{i} 4_{+} 1_{+} 0 6 -40 1^{2} 3_{i} −2 −2 1^{3} 4_{+} ∗  1^{2} 1_{+} 17 17  3 2 2 3  3 3
S_{3}  3 2 3 2 2  3 3

Discriminant class: −17

∞ 2 3 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 17 5 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
10. 1_{k} 2_{i} 4_{+} 1_{+} 0 6 -28 1^{2} 2_{i} −1 −1 1^{3} 4_{+} 1  1^{2} 1_{+} 17 17 2  3 3 2 3 2 3
S_{3} 3  2 2 2 2 3 2 2 2

Discriminant class: −6

∞ 2 3 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 17 5 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
11. 1_{k} 3_{−} 1_{k} 2_{+} -1 -11 39 1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 3 3 1^{3} 2_{+} ∗ 3 3 2 2 2 3 3
S_{3}  2 2 2 2 3 3 2 2
12. 1_{k} 3_{−} 3_{k} 2_{+} 0 51 34 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  3 3 3 2 2 2 3 3
S_{3} 3  2 2 2 2  3 2 2
13. 1_{k} 3_{−} 3_{k} 2_{+} 0 -51 -170 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  3  2 2 2 3
S_{3} 3  2 2 2 2 3 3 2 2

Discriminant class: −3

∞ 2 3 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 17 5 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. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -34 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  2  2  3 2 2 3
S_{3} 3  3 2 3 2 2 2  3
15. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -68 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  2 3 2 3  2 2 3
S_{3} 3  2 3 2 2 2 3
16. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -612 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 2 3 2 3 2 2
S_{3} 6  3 2 3 2 2 2 3
17. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -102 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 3 2 3  2 2 3
S_{3} 3  3 2  2 2 2  3
18. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -306 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 3 2 3 3 2 2
S_{3} 3  2 3 2 2 2  3
19. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -204 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2  2 3 3 2 2 3
S_{3} 12  3 2  2 2 2 3

Discriminant class: 3

∞ 2 3 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 17 5 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
20. 0_{+} 2_{i} 3_{k} 2_{+} 0 -51 34 1^{2} 2_{i} −∗ −∗ 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  2 2 3 3 2  2 2
S_{3}  3 2 2 3 2  2

Discriminant class: 6

∞ 2 3 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 17 5 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
21. 0_{+} 3_{k} 3_{i} 2_{+} 0 -51 -102 1^{2} 3_{k} −2∗ −2∗ 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  3 2 2 2  3  2
S_{3}  2 2 3 3 3 2 2 3
22. 0_{+} 3_{k} 5_{i} 2_{+} 0 -153 -714 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  3 2 2 2 3  3 2
S_{3}  2 2 3  3 2 2 3
23. 0_{+} 3_{k} 5_{i} 2_{+} 0 -153 102 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 2 2 3 3 3 2
S_{3}  2 2 3 3  2 2

Discriminant class: 51

∞ 2 3 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 17  5 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
24. 0_{+} 2_{i} 1_{i} 1_{−} 0 -51  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{−} 17∗ 17∗  2  2 2
C_{2} 2  2  2  2  2 2
25. 0_{+} 2_{i} 5_{i} 1_{−} 0 -27 -48 1^{2} 2_{i} −∗ −∗ 1^{3} 5_{i} 3∗  1^{2} 1_{−} 17∗ 17∗ 3 3 2 3 2 2
S_{3}  2 3 2  2 3 2 2

Discriminant class: 102

∞ 2 3 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 17  5 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
26. 0_{+} 3_{k} 1_{k} 1_{−} 0 -102  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{k} 3 3 1^{2} 1_{−} 17∗ 17∗ 2  2  2 2
C_{2} 2  2