Low Degree Fields Unramified Outside of {∞, 7, 13, 19}

Discriminant class: −247

∞ 7 13 19 a_{1} a_{2} a_{3} Over p = 7  Over p = 13  Over p = 19 2 3 5 11 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} 0_{+} 1_{−} 1_{k} 0 247  2 0_{+} ∗ ∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19  2 2 2  2
C_{2} 6  2 2
2. 1_{k} 0_{+} 1_{−} 1_{k} 0 1 -3 2 0_{+} ∗ ∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19 3 2 2 2 3 3 2 3
S_{3}  3 3 3 2 2 3 3 3

Discriminant class: −91

∞ 7 13 19 a_{1} a_{2} a_{3} Over p = 7  Over p = 13  Over p = 19 2 3 5 11 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} 1_{k} 1_{−} 0_{+} 0 91  1^{2} 1_{k} 7 7 1^{2} 1_{−} 13∗ 13∗  2 2  2 2
C_{2} 2  2  2 2
4. 1_{k} 1_{k} 1_{−} 2_{+} -1 13 26 1^{2} 1_{k} 7 7 1^{2} 1_{−} 13∗ 13∗ 1^{3} 2_{+} 1  2 2 3 2 2 3 3 3
S_{3} 3 2 3 3 3 2 2

Discriminant class: −19

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

Discriminant class: −7

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

Discriminant class: 1

∞ 7 13 19 a_{1} a_{2} a_{3} Over p = 7 Over p = 13 Over p = 19 2 3 5 11 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
8. 0_{+} 2_{+} 0_{+} 0_{+} -1 -2  1 1^{3} 2_{+} 1  3 0_{+} 1  3 3 3 3 3 3  3
C_{3}  3  3 3 3 3 3
9. 0_{+} 0_{+} 2_{+} 0_{+} -1 -4  -1 3 0_{+} 1  1^{3} 2_{+} 1  3 0_{+} 1  3 3  3 3 3 3
C_{3}  3 3 3  3 3 3
10. 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

Discriminant class: 1 (continued)

∞ 7 13 19 a_{1} a_{2}  a_{3} Over p = 7  Over p = 13 Over p = 19 2 3 5 11 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
11. 0_{+} 2_{+} 2_{+} 0_{+} -1 -30  64 1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 0_{+} 1  3 3 3  3 3
C_{3} 3  3 3 3 3  3 3
12. 0_{+} 2_{+} 2_{+} 0_{+} 1 -30  27 1^{3} 2_{+} 1  1^{3} 2_{+} 1  3  3  3 3 3 3
C_{3} 3  3 3 3 3 3 3
13. 0_{+} 2_{+} 0_{+} 2_{+} -1 -44 -69 1^{3} 2_{+} 1  3 0_{+} 1  1^{3} 2_{+} 1  3  3 3  3 3
C_{3} 3  3 3 3  3  3
14. 0_{+} 2_{+} 0_{+} 2_{+} 1 -44 -64 1^{3} 2_{+} 1  3 0_{+} 1  1^{3} 2_{+} 1  3  3 3 3 3 3
C_{3} 3  3 3 3 3  3 3
15. 0_{+} 0_{+} 2_{+} 2_{+} -1 -82 311 3 0_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3 3
C_{3} 3  3 3  3 3  3
16. 0_{+} 0_{+} 2_{+} 2_{+} -1 -82  64 3 0_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3
C_{3} 3  3  3 3 3 3  3
17. 0_{+} 2_{+} 2_{+} 2_{+} -1 -576 -5123 1^{3} 2_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3 3 3 3
C_{3} 9  3  3 3 3
18. 0_{+} 2_{+} 2_{+} 2_{+} -1 -576 5251 1^{3} 2_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3  3 3
C_{3} 9  3  3 3  3 3
19. 0_{+} 2_{+} 2_{+} 2_{+} -1 -576 -1665 1^{3} 2_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3  3 3 3  3
C_{3} 9  3  3  3  3
20. 0_{+} 2_{+} 2_{+} 2_{+} -1 -576  64 1^{3} 2_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3  3 3  3
C_{3} 9  3 3  3 3 3 3

Discriminant class: 13

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

Discriminant class: 133

∞ 7 13 19 a_{1} a_{2} a_{3}  Over p = 7  Over p = 13  Over p = 19 2 3 5 11 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
22. 0_{+} 1_{i} 0_{+} 1_{k} 0 -133  1^{2} 1_{i} 7∗ 7∗  1^{2} 1_{k} 19 19 2  2  2  2
C_{2}  2  2 2  2 2

Discriminant class: 1729

∞ 7 13 19 a_{1}  a_{2} a_{3} Over p = 7  Over p = 13  Over p = 19  2 3 5 11 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
23. 0_{+} 1_{k} 1_{+} 1_{i} 0 -1729  1^{2} 1_{k} 7 7 1^{2} 1_{+} 13 13 1^{2} 1_{i} 19∗ 19∗  2 2  2 2
C_{2} 2  2  2 2 2