Low Degree Fields Unramified Outside of {∞, 3, 19, 23}

Discriminant class: −1311

∞ 3 19 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 19 Over p = 23 2 5 7 11 13 17 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_{k} 1_{k} 0 1311  1^{2} 1_{k} 3 3 1^{2} 1_{k} 19 19 1^{2} 1_{k} 23 23  2  2  2
C_{2} 28  2 2 2  2
2. 1_{k} 3_{k} 1_{k} 1_{k} 0  -3 -21 1^{3} 3_{k} 3  1^{2} 1_{k} 19 19 1^{2} 1_{k} 23 23 3 3 2 3 2 3 3 2
S_{3}  3 3 2 2 2  2 3

Discriminant class: −23

∞ 3 19 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 19  Over p = 23  2 5 7 11 13 17 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} 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
4. 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

Discriminant class: −19

∞ 3 19 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 19  Over p = 23 2 5 7 11 13 17 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_{+} 1_{i} 0_{+} -1 5  2 0_{+} ∗ ∗ 1^{2} 1_{i} 19∗ 19∗  2  2  2 2
C_{2}  2 2  2 2  2
6. 1_{k} 4_{+} 1_{i} 0_{+} 0 6 -5 1^{3} 4_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 3 0_{+} 1  2  3 3 2 3 2 2
S_{3}  2 2 3 3 2 2 3 2

Discriminant class: −3

∞ 3 19 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 19 Over p = 23 2 5 7 11 13 17 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
7. 1_{k} 1_{i} 0_{+} 0_{+} 1 1  1^{2} 1_{i} 3∗ 3∗  2 0_{+} ∗ ∗ 2 2  2  2 2
C_{2}  2  2 2 2
8. 1_{k} 5_{i} 0_{+} 0_{+} 0 0  -3 1^{3} 5_{i} 3∗  3 0_{+} 1  2 0_{+} ∗ ∗ 2 2 3 2 3 2 2 3
S_{3}  3 2 3 2 2 2
9. 1_{k} 1_{i} 2_{+} 0_{+} -1 -6 -12 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1 2 0_{+} ∗ ∗ 2 2 3 2 3 2 2 3
S_{3} 3  3 2 3 2 2 2 3 3
10. 1_{k} 3_{i} 0_{+} 2_{+} 0 0  -23 1^{3} 3_{i} 3∗  3 0_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 2 2
S_{3}  2 3 2 2 2  3
11. 1_{k} 5_{i} 2_{+} 0_{+} 0 0 -171 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 0_{+} ∗ ∗ 2 2 3 2 3 2 2
S_{3} 6  2  2 2 2 3 3
12. 1_{k} 5_{i} 2_{+} 0_{+} 0 0  -57 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 0_{+} ∗ ∗ 2 2  2  2 2 3
S_{3} 6  3 2 3 2 2 2 3 3
13. 1_{k} 5_{i} 0_{+} 2_{+} 0 0 -207 1^{3} 5_{i} 3∗  3 0_{+} 1  1^{3} 2_{+} ∗  2 2 3 2  2 2 3
S_{3} 8  3 2  2 2 2  3

Discriminant class: −3 (continued)

∞ 3 19 23 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 19 Over p = 23 2 5 7 11 13 17 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} 0_{+} 2_{+} 0 0  -69 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 2  2 3 2 2 3
S_{3}  3 2 3 2 2 2  3
15. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -437 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2  2 2 3
S_{3} 6  3 2 3 2 2 2 3 3
16. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -8303 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 3 2 2 3
S_{3} 6  3 2  2 2 2 3
17. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -3933 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 3 2 2
S_{3} 3  2 3 2 2 2 3 3
18. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1311 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 2 2 3
S_{3} 9  3 2  2 2 2 3 3
19. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -24909 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2  2 2
S_{3} 54  2 3 2 2 2 3
20. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -30153 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: 1

∞ 3 19 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 19 Over p = 23 2 5 7 11 13 17 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_{+} 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
22. 0_{+} 0_{+} 2_{+} 0_{+} -1 -6 7 3 0_{+} 1  1^{3} 2_{+} 1  3 0_{+} 1  3 3  3 3 3
C_{3}  3 3 3 3 3 3 3
23. 0_{+} 4_{+} 2_{+} 0_{+} 0 -57 19 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 0_{+} 1  3  3 3 3 3  3
C_{3} 3  3  3  3
24. 0_{+} 4_{+} 2_{+} 0_{+} 0 -57 152 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 3 3  3 3 3
C_{3} 3  3  3 3 3 3

Discriminant class: 57

∞ 3 19 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 19  Over p = 23 2 5 7 11 13 17 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_{+} 1_{k} 1_{i} 0_{+} 0 -57  1^{2} 1_{k} 3 3 1^{2} 1_{i} 19∗ 19∗ 2 0_{+} ∗ ∗  2  2 2 2  2
C_{2}  2  2  2
26. 0_{+} 3_{k} 1_{i} 2_{+} 0 -69 -92 1^{3} 3_{k} 3  1^{2} 1_{i} 19∗ 19∗ 1^{3} 2_{+} ∗  2 3 2 2 2  2
S_{3}  2 3 3 2  3 3 2

Discriminant class: 69

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

Discriminant class: 69 (continued)

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

Discriminant class: 437

∞ 3 19 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 19  Over p = 23  2 5 7 11 13 17 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_{+} 0_{+} 1_{k} 1_{i} 0 -437  2 0_{+} ∗ ∗ 1^{2} 1_{k} 19 19 1^{2} 1_{i} 23∗ 23∗ 2 2 2 2 2 2 2 2
C_{2}  2 2  2 2