Low Degree Fields Unramified Outside of {∞, 2, 3, 23}

Discriminant class: −138

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23  5 7 11 13 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} 3_{k} 1_{i} 1_{i} 0 138  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{i} 23∗ 23∗ 2  2  2  2
C_{2} 8  2 2  2 2  2
2. 1_{k} 3_{k} 3_{i} 1_{i} 0 15 -34 1^{2} 3_{k} −2∗ −2∗ 1^{3} 3_{i} 3∗  1^{2} 1_{i} 23∗ 23∗ 2 3 3 2  2 3 2
S_{3}  2 2 3 2 2 3 2
3. 1_{k} 3_{k} 5_{i} 1_{i} 0 63 -66 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗  1^{2} 1_{i} 23∗ 23∗ 2 3  2  2 3 2
S_{3}  3 2 2  2 2 3 2
4. 1_{k} 3_{k} 5_{i} 1_{i} 0 45 -114 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗  1^{2} 1_{i} 23∗ 23∗ 2 3 3 2  2  2
S_{3} 4  3 2 2 3 2 2 3 2
5. 1_{k} 3_{k} 5_{i} 1_{i} 0 -9 -42 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗  1^{2} 1_{i} 23∗ 23∗ 2  3 2  2 3 2
S_{3}  3 2 2 3 2 2  2

Discriminant class: −69

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23  5 7 11 13 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
6. 1_{k} 2_{i} 1_{k} 1_{i} 0 69  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{k} 3 3 1^{2} 1_{i} 23∗ 23∗  2  2 2
C_{2} 8  2 2  2
7. 1_{k} 2_{i} 3_{k} 1_{i} 0 -6 -20 1^{2} 2_{i} −∗ −∗ 1^{3} 3_{k} 3  1^{2} 1_{i} 23∗ 23∗  3 2 3 3 3 2 2
S_{3}  2 2 3 3 3 3 2

Discriminant class: −46

∞ 2 3 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 23  5 7 11 13 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
8. 1_{k} 3_{+} 0_{+} 1_{i} 0 46  1^{2} 3_{+} 2 2 2 0_{+} ∗ ∗ 1^{2} 1_{i} 23∗ 23∗  2  2 2  2
C_{2} 4  2
9. 1_{k} 3_{+} 4_{+} 1_{i} 0 -3 -94 1^{2} 3_{+} 2 2 1^{3} 4_{+} ∗  1^{2} 1_{i} 23∗ 23∗ 3 2 3 2 2  2 3
S_{3}  3 3  3 2 3 3

Discriminant class: −23

∞ 2 3 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 23  5 7 11 13 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
10. 1_{k} 0_{+} 0_{+} 1_{i} 0 23  1^{2} 1_{i} 23∗ 23∗ 2 2 2  2 2
C_{2} 3  2  2  2  2 2
11. 1_{k} 0_{+} 0_{+} 1_{i} -1 0 1 3 0_{+} 1  3 0_{+} 1  1^{2} 1_{i} 23∗ 23∗ 2 2 2 3 2 2 3 3
S_{3}  2 3 2 3 2  2 2

Discriminant class: −6

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23 5 7 11 13 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
12. 1_{k} 3_{−} 1_{k} 0_{+} 0 6  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 3 3 2 0_{+} ∗ ∗  2 2 2
C_{2} 2  2 2 2 2  2 2
13. 1_{k} 3_{−} 3_{k} 0_{+} 0 3 -2 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{k} 3  2 0_{+} ∗ ∗ 3 3 3 2 2 2  3
S_{3}  2 2 2 2 3  2 2
14. 1_{k} 3_{−} 1_{k} 2_{+} -1 8 40 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
15. 1_{k} 3_{−} 3_{k} 2_{+} 0 69 -138 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{k} 3 1^{3} 2_{+} ∗ 3 3 3 2 2 2 3
S_{3} 3  2 2 2 2 3 3 2 2
16. 1_{k} 3_{−} 3_{k} 2_{+} 0 69 -322 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  3  2 2 2 3 3
S_{3} 3  2 2 2 2  3 2 2

Discriminant class: −3

∞ 2 3 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 23 5 7 11 13 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
17. 1_{k} 0_{+} 1_{i} 0_{+} 1 1  2 0_{+} ∗ ∗ 1^{2} 1_{i} 3∗ 3∗ 2 0_{+} ∗ ∗ 2  2  2  2
C_{2}  2  2 2 2
18. 1_{k} 2_{+} 3_{i} 0_{+} 0 0 -2 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  2 0_{+} ∗ ∗ 2 3 2 3 2 3 2
S_{3}  3 2  2 2 2 3 3
19. 1_{k} 0_{+} 5_{i} 0_{+} 0 0 -3 2 0_{+} ∗ ∗ 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 2 3 2 3 2 3 2 3
S_{3}  3 2 3 2 2 2
20. 1_{k} 2_{+} 5_{i} 0_{+} 0 0 -12 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 2 0_{+} ∗ ∗ 2 3 2 2 2 3
S_{3}  3 2 3 2 2 2 3 3
21. 1_{k} 2_{+} 5_{i} 0_{+} 0 0 -6 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 2 0_{+} ∗ ∗ 2 2 3 2 3 2 3
S_{3}  2 3 2 2 2 3 3
22. 1_{k} 2_{+} 1_{i} 2_{+} 0 0 -46 1^{3} 2_{+} ∗ 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗ 2 3 2 3 2 2
S_{3}  3 2 3 2 2 2 3 3
23. 1_{k} 0_{+} 3_{i} 2_{+} 0 0 -23 2 0_{+} ∗ ∗ 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  2 3 2 3 2 3 2
S_{3}  2 3 2 2 2  3
24. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -92 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  2  2  2 3 2
S_{3} 3  3 2 3 2 2 2 3
25. 1_{k} 0_{+} 5_{i} 2_{+} 0 0 -69 2 0_{+} ∗ ∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2  2 3 2  2 3
S_{3}  3 2 3 2 2 2  3
26. 1_{k} 0_{+} 5_{i} 2_{+} 0 0 -207 2 0_{+} ∗ ∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 3 2  2 3 2 3
S_{3} 8  3 2  2 2 2  3
27. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -276 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 3 2 3 2 3 2 3
S_{3} 3  3 2 3 2 2 2 3
28. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -138 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 2 3 2 2 3 2 3
S_{3} 3  2 3 2 2 2 3 3
29. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -414 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2  2 3 2 3 2 3
S_{3} 6  3 2  2 2 2 3 3
30. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -828 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 3 2 3 2  2 3
S_{3} 6  2  2 2 2 3

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23 5 7 11 13 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
31. 1_{k} 3_{i} 0_{+} 0_{+} 0 2  1^{2} 3_{i} −2 −2  2 0_{+} ∗ ∗ 2 2  2  2 2
C_{2}  2  2 2  2
32. 1_{k} 3_{i} 4_{+} 0_{+} 0 -3 -10 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  2 0_{+} ∗ ∗ 2 2 3 2 3 3 2 2
S_{3} 3  2 3 3 2 2 3 2 3
33. 1_{k} 3_{i} 0_{+} 2_{+} -1 -15 -29 1^{2} 3_{i} −2 −2 3 0_{+} 1  1^{3} 2_{+} ∗  2 2  2 3 3 2 2
S_{3}  2  3 2 2 3 2 3
34. 1_{k} 3_{i} 4_{+} 2_{+} 0 69 -46 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 3 2 2
S_{3} 6  2 3  2 2 3 2 3
35. 1_{k} 3_{i} 4_{+} 2_{+} 0 69 -874 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  2 2 3 2  2 2
S_{3} 3  2 3 3 2 2  2

Discriminant class: −1

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23 5 7 11 13 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
36. 1_{k} 2_{i} 0_{+} 0_{+} 0 1  1^{2} 2_{i} −1 −1 2 0_{+} ∗ ∗ 2 0_{+} ∗ ∗  2 2  2  2
C_{2}  2 2  2  2
37. 1_{k} 2_{i} 4_{+} 0_{+} 0 -3 -4 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗  2 0_{+} ∗ ∗ 3 2 2 3 3 2 3 2
S_{3}  3  2 2  2 3 2
38. 1_{k} 2_{i} 0_{+} 2_{+} -1 8 -6 1^{2} 2_{i} −1 −1 2 0_{+} ∗ ∗ 1^{3} 2_{+} ∗  3 2 2 3  2 3 2
S_{3} 2  3 2 2 3 2 3 2
39. 1_{k} 2_{i} 4_{+} 2_{+} 0 69 -598 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗ 1^{3} 2_{+} ∗ 3 2 2 3 2 3 2
S_{3} 3  3 3 2 2 3 2 3 2
40. 1_{k} 2_{i} 4_{+} 2_{+} 0 69 -230 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗  1^{3} 2_{+} ∗  2 2 3 3 2  2
S_{3} 6  3 3 2 2 3 2  2

Discriminant class: 1

∞ 2 3 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 23 5 7 11 13 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
41. 0_{+} 0_{+} 4_{+} 0_{+} 0 -3 1 3 0_{+} 1  1^{3} 4_{+} 1  3 0_{+} 1  3 3 3 3  3 3
C_{3}  3 3 3  3 3 3

Discriminant class: 2

∞ 2 3 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 23 5 7 11 13 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
42. 0_{+} 3_{+} 0_{+} 0_{+} 0 -2  1^{2} 3_{+} 2 2 2 0_{+} ∗ ∗  2  2 2  2 2
C_{2}  2  2  2 2 2 2

Discriminant class: 3

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23 5 7 11 13 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
43. 0_{+} 2_{i} 1_{k} 0_{+} 0 -3  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{k} 3 3  2 2  2 2 2 2
C_{2}  2 2  2  2

Discriminant class: 6

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23 5 7 11 13 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
44. 0_{+} 3_{k} 1_{i} 0_{+} 0 -6  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{i} 3∗ 3∗  2 2 2 2  2
C_{2}  2 2  2 2
45. 0_{+} 3_{k} 5_{i} 0_{+} 0 -9 -6 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗  3 0_{+} 1  3 2 2 2 2 3 3 2
S_{3}  2 2  3 3 2 2 3

Discriminant class: 23

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23 5 7 11 13 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
46. 0_{+} 2_{i} 0_{+} 1_{k} 0 -23  1^{2} 2_{i} −1 −1 2 0_{+} ∗ ∗ 1^{2} 1_{k} 23 23 2  2  2
C_{2}  2  2 2 2 2

Discriminant class: 46

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23 5 7 11 13 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
47. 0_{+} 3_{i} 0_{+} 1_{k} 0 -46  1^{2} 3_{i} −2 −2  1^{2} 1_{k} 23 23  2 2 2 2 2 2
C_{2}  2 2  2

Discriminant class: 69

∞ 2 3 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 23 5 7 11 13 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
48. 0_{+} 0_{+} 1_{i} 1_{k} 0 -69  2 0_{+} ∗ ∗ 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{k} 23 23  2  2 2
C_{2}  2 2 2 2  2 2 2
49. 0_{+} 0_{+} 3_{i} 1_{k} 0 -6 -3 2 0_{+} ∗ ∗ 1^{3} 3_{i} 3∗  1^{2} 1_{k} 23 23 3 2 3 3 3 2 2 3
S_{3}  2 2 2 2 3 2 2 2
50. 0_{+} 2_{+} 5_{i} 1_{k} 0 -36 60 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 1^{2} 1_{k} 23 23 2 3 3 3 2 2
S_{3}  2 2 2 2 3 2 2 2
51. 0_{+} 2_{+} 5_{i} 1_{k} 0 -18 -6 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 1^{2} 1_{k} 23 23 3 2 3 2 2 3
S_{3}  2 2 2 2 3 2 2 2
52. 0_{+} 2_{+} 5_{i} 1_{k} 0 -36 -78 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{2} 1_{k} 23 23 3 2 3  3 2 2 3
S_{3}  2 2 2 2  2 2 2

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23 5 7 11 13 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
53. 0_{+} 3_{−} 1_{k} 1_{k} 0 -138  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 3 3 1^{2} 1_{k} 23 23 2 2 2 2
C_{2} 2  2  2 2