Low Degree Fields Unramified Outside of {∞, 3, 11, 19}

Discriminant class: −627

∞ 3 11 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 11 Over p = 19 2 5 7 13 17 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} 1_{k} 1_{k} 1_{k} 0 627  1^{2} 1_{k} 3 3 1^{2} 1_{k} 11 11 1^{2} 1_{k} 19 19 2 2 2  2 2 2
C_{2} 4  2 2 2 2  2 2
2. 1_{k} 3_{k} 1_{k} 1_{k} 0 6 -43 1^{3} 3_{k} 3  1^{2} 1_{k} 11 11 1^{2} 1_{k} 19 19 2 2 2 3 3 2 2 2
S_{3} 2 2 2 2 3 2 2

Discriminant class: −19

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

Discriminant class: −11

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

Discriminant class: −3

∞ 3 11 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 11 Over p = 19 2 5 7 13 17 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
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∗  2 0_{+} ∗ ∗ 3 0_{+} 1  2 2 3 3 2 2 2 3
S_{3}  3 2 3 2 2 2
9. 1_{k} 1_{i} 0_{+} 2_{+} -1 -6 -12 1^{2} 1_{i} 3∗ 3∗ 2 0_{+} ∗ ∗ 1^{3} 2_{+} 1 2 2 3 3 2 2 2 3
S_{3} 3  3 2 3 2 2 2 3 3
10. 1_{k} 3_{i} 2_{+} 0_{+} 0 0  -11 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  2 2 3 3 2 2 2 3
S_{3} 2  2  2 2 2  3
11. 1_{k} 5_{i} 2_{+} 0_{+} 0 0  -99 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  3 0_{+} 1  2 2  2 2 2 3
S_{3}  3 2 3 2 2 2  3
12. 1_{k} 5_{i} 2_{+} 0_{+} 0 0  -33 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  3 0_{+} 1  2 2 3 3 2 2 2
S_{3}  3 2 3 2 2 2  3
13. 1_{k} 5_{i} 0_{+} 2_{+} 0 0  -57 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 1^{3} 2_{+} 1  2 2  2 2 2 3
S_{3} 6  3 2 3 2 2 2 3 3

Discriminant class: −3 (continued)

∞ 3 11 19 a_{1} a_{2}  a_{3}  Over p = 3  Over p = 11 Over p = 19 2 5 7 13 17 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} 5_{i} 0_{+} 2_{+} 0 0 -171 1^{3} 5_{i} 3∗  2 0_{+} ∗ ∗ 1^{3} 2_{+} 1  2 2 3 3 2 2 2
S_{3} 6  2  2 2 2 3 3
15. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -209 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2  2 2 2
S_{3} 27  3 2 3 2 2 2 3
16. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -2299 1^{3} 3_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 2 3 3 2 2 2 3
S_{3} 9  3 2 3 2 2 2 3 3
17. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -627 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 2 3 3 2 2 2 3
S_{3} 12  3 2 3 2 2 2 3
18. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -6897 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 2 3 3 2 2 2
S_{3} 3  3 2 3 2 2 2 3 3
19. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -11913 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2  2 2 2 3
S_{3} 3  2  2 2 2 3 3
20. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1881 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 3 3 2 2 2 3
S_{3} 6  2  2 2 2 3

Discriminant class: 1

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

Discriminant class: 33

∞ 3 11 19 a_{1} a_{2} a_{3}  Over p = 3  Over p = 11  Over p = 19 2 5 7 13 17 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
25. 0_{+} 1_{i} 1_{k} 0_{+} 0 -33 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{k} 11 11 2 0_{+} ∗ ∗ 2 2 2 2
C_{2}  2 2 2 2 2
26. 0_{+} 5_{i} 1_{k} 0_{+} 0 -9 -3 1^{3} 5_{i} 3∗  1^{2} 1_{k} 11 11 2 0_{+} ∗ ∗ 3 2 2 2 3 2 3
S_{3}  3 3 2 2 2 2 2

Discriminant class: 57

∞ 3 11 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 11  Over p = 19  2 5 7 13 17 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
27. 0_{+} 1_{k} 0_{+} 1_{i} 0 -57  1^{2} 1_{k} 3 3 2 0_{+} ∗ ∗ 1^{2} 1_{i} 19∗ 19∗  2  2 2 2  2
C_{2}  2  2  2

Discriminant class: 57 (continued)

∞ 3 11 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 11  Over p = 19  2 5 7 13 17 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
28. 0_{+} 3_{k} 2_{+} 1_{i} 0 -33 -55 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 3 2 3 2 2 2  2
S_{3}  2  3 2 3 3 3 2

Discriminant class: 209

∞ 3 11 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 11  Over p = 19 2 5 7 13 17 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
29. 0_{+} 0_{+} 1_{i} 1_{k} 0 -209  2 0_{+} ∗ ∗ 1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{k} 19 19  2  2  2
C_{2}  2  2  2 2 2 2