Low Degree Fields Unramified Outside of {∞, 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} 0_{+} 1_{−} 1_{k} 0 391  2 0_{+} ∗ ∗ 1^{2} 1_{−} 17∗ 17∗ 1^{2} 1_{k} 23 23  2 2 2
C_{2} 14  2 2  2  2
2. 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: −51

∞ 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
3. 1_{k} 1_{k} 1_{−} 0_{+} 0 51  1^{2} 1_{k} 3 3 1^{2} 1_{−} 17∗ 17∗  2  2  2
C_{2} 2  2  2 2 2 2
4. 1_{k} 3_{k} 1_{−} 0_{+} 0 3 -8 1^{3} 3_{k} 3  1^{2} 1_{−} 17∗ 17∗ 3 0_{+} 1  2 3 2  3 3 3 2
S_{3}  2 3 3 2 2 2 2 3

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
5. 1_{k} 0_{+} 0_{+} 1_{i} 0 23  2 0_{+} ∗ ∗ 1^{2} 1_{i} 23∗ 23∗  2 2 2  2
C_{2} 3  2  2  2  2 2
6. 1_{k} 0_{+} 0_{+} 1_{i} -1 0  1 3 0_{+} 1  2 0_{+} ∗ ∗ 1^{2} 1_{i} 23∗ 23∗ 3 2 2 2 3 2 3 3
S_{3}  2 3 2 3 2  2 2
7. 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
8. 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
9. 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

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
10. 1_{k} 1_{i} 0_{+} 0_{+} 1 1  1^{2} 1_{i} 3∗ 3∗ 2 0_{+} ∗ ∗ 2 0_{+} ∗ ∗ 2 2  2  2
C_{2}  2  2 2 2
11. 1_{k} 5_{i} 0_{+} 0_{+} 0 0  -3 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 2 0_{+} ∗ ∗ 2 2 3 2 3 3 2 3
S_{3}  3 2 3 2 2 2
12. 1_{k} 1_{i} 2_{+} 0_{+} -1 6 -12 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗ 2 0_{+} ∗ ∗ 2 2 3 2 3 3 2 3
S_{3}  3 2 3 2 2 2 3 3
13. 1_{k} 3_{i} 0_{+} 2_{+} 0 0  -23 1^{3} 3_{i} 3∗  2 0_{+} ∗ ∗ 1^{3} 2_{+} ∗  2 2 3 2 3 3 2
S_{3}  2 3 2 2 2  3

Discriminant class: −3 (continued)

∞ 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
14. 1_{k} 5_{i} 2_{+} 0_{+} 0 0  -51 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 0_{+} ∗ ∗ 2 2 3 2  3 2 3
S_{3} 3  2  2 2 2 3 3
15. 1_{k} 5_{i} 2_{+} 0_{+} 0 0 -153 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 0_{+} ∗ ∗ 2 2  2 3  2
S_{3} 9  3 2 3 2 2 2 3 3
16. 1_{k} 5_{i} 0_{+} 2_{+} 0 0 -207 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 1^{3} 2_{+} ∗  2 2 3 2  3 2 3
S_{3} 8  3 2  2 2 2  3
17. 1_{k} 5_{i} 0_{+} 2_{+} 0 0  -69 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 1^{3} 2_{+} ∗  2 2  2 3  2 3
S_{3}  3 2 3 2 2 2  3
18. 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
19. 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
20. 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
21. 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
22. 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
23. 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

Discriminant class: 1

∞ 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
24. 0_{+} 4_{+} 0_{+} 0_{+} 0 -3 1 1^{3} 4_{+} 1  3 0_{+} 1  3 3 3 3 3  3 3
C_{3}  3 3 3  3 3 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
25. 0_{+} 0_{+} 1_{+} 0_{+} 1 -4  2 0_{+} ∗ ∗ 1^{2} 1_{+} 17 17 2 0_{+} ∗ ∗  2 2 2  2 2
C_{2}  2 2  2
26. 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: 69

∞ 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
27. 0_{+} 1_{i} 0_{+} 1_{k} 0 -69  1^{2} 1_{i} 3∗ 3∗  1^{2} 1_{k} 23 23 2  2  2 2
C_{2}  2 2 2 2  2 2 2

Discriminant class: 69 (continued)

∞ 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
28. 0_{+} 3_{i} 0_{+} 1_{k} 0 -6 -3 1^{3} 3_{i} 3∗ 3 0_{+} 1 1^{2} 1_{k} 23 23 2 3 2 3 3 2 2 3
S_{3}  2 2 2 2 3 2 2 2

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
29. 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