Low Degree Fields Unramified Outside of {∞, 3, 23, 29}

Discriminant class: −667

∞ 3 23 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 23  Over p = 29 2 5 7 11 13 17 19 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_{i} 1_{+} 0 667  2 0_{+} ∗ ∗ 1^{2} 1_{i} 23∗ 23∗ 1^{2} 1_{+} 29 29 2 2 2  2
C_{2} 4  2  2 2  2
2. 1_{k} 4_{+} 1_{i} 1_{+} 0 -39 -202 1^{3} 4_{+} ∗  1^{2} 1_{i} 23∗ 23∗ 1^{2} 1_{+} 29 29 2 2 2  3  3 2
S_{3}  3 2  2 2 3 3 2

Discriminant class: −87

∞ 3 23 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 23  Over p = 29  2 5 7 11 13 17 19 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} 0_{+} 1_{−} 0 87  1^{2} 1_{k} 3 3 2 0_{+} ∗ ∗ 1^{2} 1_{−} 29∗ 29∗  2  2 2
C_{2} 6  2  2  2 2 2
4. 1_{k} 1_{k} 0_{+} 1_{−} -1 2  1 1^{2} 1_{k} 3 3 2 0_{+} ∗ ∗ 1^{2} 1_{−} 29∗ 29∗ 3 2 3 3 3 3 2 2
S_{3}  2  2 3 2 2 2 3
5. 1_{k} 3_{k} 2_{+} 1_{−} 0 69 966 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{−} 29∗ 29∗  2  3 3 3 2 2
S_{3} 3  2  2 3 2 2 2
6. 1_{k} 3_{k} 2_{+} 1_{−} 0 69 -299 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{−} 29∗ 29∗ 3 2 3 3  2 2
S_{3} 3  2  2  2 2 2 3
7. 1_{k} 3_{k} 2_{+} 1_{−} 0 -69 -253 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{−} 29∗ 29∗ 3 2 3  3 3 2 2
S_{3} 3  2  2 3 2 2 2 3

Discriminant class: −23

∞ 3 23 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 23  Over p = 29 2 5 7 11 13 17 19 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
8. 1_{k} 0_{+} 1_{i} 0_{+} 0 23  1^{2} 1_{i} 23∗ 23∗  2 2 2  2 2
C_{2} 3  2  2  2  2 2
9. 1_{k} 0_{+} 1_{i} 0_{+} -1 0 1 3 0_{+} 1  1^{2} 1_{i} 23∗ 23∗ 3 0_{+} 1  3 2 2 2 3 2 2 3
S_{3}  2 3 2 3 2  2 2

Discriminant class: −3

∞ 3 23 29 a_{1} a_{2} a_{3} Over p = 3 Over p = 23 Over p = 29 2 5 7 11 13 17 19 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 2 3 3
S_{3}  3 2 3 2 2 2
12. 1_{k} 3_{i} 2_{+} 0_{+} 0 0  -23 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  2 0_{+} ∗ ∗ 2 2 3 2 3 2 3
S_{3}  2 3 2 2 2  3
13. 1_{k} 3_{i} 0_{+} 2_{+} 0 0  -29 1^{3} 3_{i} 3∗  2 0_{+} ∗ ∗ 1^{3} 2_{+} ∗  2 2  2 3 2 3
S_{3}  2 3 2 2 2 3 3

Discriminant class: −3 (continued)

∞ 3 23 29 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 23 Over p = 29 2 5 7 11 13 17 19 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 -207 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 0_{+} ∗ ∗ 2 2 3 2  2 3 3
S_{3} 8  3 2  2 2 2  3
15. 1_{k} 5_{i} 2_{+} 0_{+} 0 0  -69 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 0_{+} ∗ ∗ 2 2  2 3 2  3
S_{3}  3 2 3 2 2 2  3
16. 1_{k} 5_{i} 0_{+} 2_{+} 0 0 -261 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 1^{3} 2_{+} ∗  2 2 3 2  2 3 3
S_{3}  3 2 3 2 2 2 3 3
17. 1_{k} 5_{i} 0_{+} 2_{+} 0 0  -87 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 1^{3} 2_{+} ∗  2 2 3 2 3 2  3
S_{3}  3 2  2 2 2 3 3
18. 1_{k} 1_{i} 2_{+} 2_{+} 0 0 -667 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2 3 2 3 2 3
S_{3} 19  2  2 2 2 3
19. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -15341 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2 3 2  2
S_{3} 3  2 3 2 2 2 3 3
20. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -58029 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2  2 3 2 3 3
S_{3} 3  3 2  2 2 2 3 3
21. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -2001 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2  2  2 3 3
S_{3} 84  3 2 3 2 2 2 3
22. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -46023 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 2 3 2 3 2 3 3
S_{3} 24  3 2 3 2 2 2 3 3
23. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -6003 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 2 3 2 3 2 3
S_{3} 3  3 2 3 2 2 2 3

Discriminant class: 1

∞ 3 23 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 23 Over p = 29 2 5 7 11 13 17 19 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 0_{+} 1  3 3 3 3 3  3
C_{3}  3 3 3  3 3 3

Discriminant class: 29

∞ 3 23 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 23  Over p = 29 2 5 7 11 13 17 19 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_{+} 0_{+} 1_{+} 0 -29  2 0_{+} ∗ ∗  1^{2} 1_{+} 29 29 2  2  2 2 2
C_{2}  2 2 2 2  2
26. 0_{+} 4_{+} 0_{+} 1_{+} 0 -12 -13 1^{3} 4_{+} ∗  3 0_{+} 1  1^{2} 1_{+} 29 29 2 3 3 2  2 2 2
S_{3}  2 2 2 2 3 3 2 3

Discriminant class: 69

∞ 3 23 29 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 23  Over p = 29 2 5 7 11 13 17 19 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} 1_{k} 0_{+} 0 -69  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{k} 23 23 2 0_{+} ∗ ∗ 2  2  2
C_{2}  2 2 2 2  2 2 2

Discriminant class: 69 (continued)

∞ 3 23 29 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 23  Over p = 29 2 5 7 11 13 17 19 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} 1_{k} 0_{+} 0 -6 -3 1^{3} 3_{i} 3∗ 1^{2} 1_{k} 23 23 2 0_{+} ∗ ∗ 2 3 2 3 3 3 2 3
S_{3}  2 2 2 2 3 2 2 2
29. 0_{+} 5_{i} 1_{k} 2_{+} 0 -522 -4089 1^{3} 5_{i} 3∗ 1^{2} 1_{k} 23 23 1^{3} 2_{+} ∗ 2 2 3 3 2 3
S_{3}  2 2 2 2 3 2 2 2
30. 0_{+} 5_{i} 1_{k} 2_{+} 0 -261 -1392 1^{3} 5_{i} 3∗  1^{2} 1_{k} 23 23 1^{3} 2_{+} ∗  2 3 2 3 3 3 2 3
S_{3}  2 2 2 2  2 2 2
31. 0_{+} 5_{i} 1_{k} 2_{+} 0 -522 -87 1^{3} 5_{i} 3∗  1^{2} 1_{k} 23 23 1^{3} 2_{+} ∗  2 3 2  3 2
S_{3}  2 2 2 2 3 2 2 2

Discriminant class: 2001

∞ 3 23 29 a_{1}  a_{2} a_{3} Over p = 3  Over p = 23  Over p = 29  2 5 7 11 13 17 19 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
32. 0_{+} 1_{k} 1_{k} 1_{−} 0 -2001  1^{2} 1_{k} 3 3 1^{2} 1_{k} 23 23 1^{2} 1_{−} 29∗ 29∗  2 2  2  2
C_{2} 2  2  2