Low Degree Fields Unramified Outside of {∞, 3, 13}

Discriminant class: −39

∞ 3 13 a_{1} a_{2} a_{3} Over p = 3 Over p = 13 2 5 7 11 17 19 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 1_{k} 1_{i} 1_{+} 0 39  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 13 13  2  2 2 2 2
C_{2} 4  2 2  2
2. 1_{k} 3_{i} 1_{+} 0 3 -3 1^{3} 3_{i} 3∗  1^{2} 1_{+} 13 13 3 3 2  2 2 2 2
S_{3}  2 2  3 3 2 3 3
3. 1_{k} 5_{i} 1_{+} 0 -9 -24 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13  3 2 3 2 2 2 2
S_{3}  2 2 3 3 3 2 3 3
4. 1_{k} 5_{i} 1_{+} 0 -9 -15 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 3  2 3 2 2 2 2
S_{3}  2 2 3  3 2  3
5. 1_{k} 5_{i} 1_{+} 0 9 -3 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 3 3 2 3 2 2 2 2
S_{3} 2 2 3 3 2 3

Discriminant class: −3

∞ 3 13 a_{1} a_{2} a_{3} Over p = 3 Over p = 13 2 5 7 11 17 19 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
6. 1_{k} 1_{i} 0_{+} 1 1  1^{2} 1_{i} 3∗ 3∗  2 2  2 2  2 2
C_{2}  2  2 2 2
7. 1_{k} 5_{i} 0_{+} 0 0 -3 1^{3} 5_{i} 3∗  3 0_{+} 1  2 2 3 2 2 3 2 2
S_{3}  3 3 2 3 2 2 2
8. 1_{k} 3_{i} 2_{+} 0 0 -13 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 2 2 2 2 3 2 2
S_{3} 3  3 3 2 3 2 2 2 3
9. 1_{k} 5_{i} 2_{+} 0 0 -117 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2 2 3 2 2
S_{3} 3  3  2 3 2 2 2 3
10. 1_{k} 5_{i} 2_{+} 0 0 -39 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2 2  2 2
S_{3} 6  3 2  2 2 2 3

Discriminant class: 1

∞ 3 13 a_{1} a_{2} a_{3} Over p = 3  Over p = 13 2 5 7 11 17 19 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
11. 0_{+} 4_{+} 0_{+} 0 -3 1 1^{3} 4_{+} 1  3 0_{+} 1  3 3 3 3  3 3
C_{3}  3  3 3 3  3 3
12. 0_{+} 0_{+} 2_{+} -1 -4 -1 3 0_{+} 1  1^{3} 2_{+} 1  3  3 3 3 3 3 3
C_{3}  3 3 3  3 3
13. 0_{+} 4_{+} 2_{+} 0 -39 91 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 3  3 3 3  3
C_{3} 3  3 3  3  3
14. 0_{+} 4_{+} 2_{+} 0 -39 -26 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 3  3 3 3
C_{3} 3  3 3 3 3 3  3

Discriminant class: 13

∞ 3 13 a_{1} a_{2} a_{3} Over p = 3  Over p = 13 2 5 7 11 17 19 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
15. 0_{+} 0_{+} 1_{+} 1 -3  1^{2} 1_{+} 13 13 2 2 2 2  2
C_{2}  2 2 2  2  2