Low Degree Fields Ramifying Above {3, 7, 13}

Discriminant class: −91

∞ 3 7 13 a_{1} a_{2} a_{3} Over p = 3  Over p = 7  Over p = 13  2 5 11 17 19 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} 4_{+} 1_{k} 1_{−} 0 -12 -23 1^{3} 4_{+} ∗  1^{2} 1_{k} 7 7 1^{2} 1_{−} 13∗ 13∗ 2 3 2 2 3  3 3
S_{3}  2 3  3  2 2

Discriminant class: −3

∞ 3 7 13 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 13 2 5 11 17 19 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
2. 1_{k} 1_{i} 2_{+} 2_{+} 0 0 -91 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} 1 2 2 2 2 3 2 2
S_{3} 9  3 2 3 2 2 2 3
3. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -637 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2 2 3 2 2 3
S_{3} 9  2  2 2 2  3
4. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -3549 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2 2 3 2 2
S_{3} 9  3 2 3 2 2 2  3
5. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1911 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2 2  2 2 3
S_{3} 45  3 2 3 2 2 2  3
6. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -273 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2 2  2 2 3
S_{3} 27  2 3 2 2 2 3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -819 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 2 2 3 2 2 3
S_{3} 9  3 2  2 2 2 3

Discriminant class: 1

∞ 3 7 13 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 13 2 5 11 17 19 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_{+} 4_{+} 2_{+} 2_{+} 0 -273 728 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3  3 3 3
C_{3} 9  3  3 3  3 3
9. 0_{+} 4_{+} 2_{+} 2_{+} 0 -273 -91 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3  3 3 3
C_{3} 9  3 3 3 3
10. 0_{+} 4_{+} 2_{+} 2_{+} 0 -273 -1729 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3 3
C_{3} 9  3 3 3  3 3 3 3
11. 0_{+} 4_{+} 2_{+} 2_{+} 0 -273 1547 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3  3  3 3  3
C_{3} 9  3 3 3 3 3

Discriminant class: 273

∞ 3 7 13 a_{1} a_{2} a_{3} Over p = 3  Over p = 7  Over p = 13  2 5 11 17 19 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
12. 0_{+} 1_{k} 1_{k} 1_{−} 0 -273  1^{2} 1_{k} 3 3 1^{2} 1_{k} 7 7 1^{2} 1_{−} 13∗ 13∗  2  2 2
C_{2} 2  2 2  2 2 2 2 2