Low Degree Fields Ramifying Above {3, 5, 13}

Discriminant class: −195

∞ 3 5 13 a_{1} a_{2} a_{3} Over p = 3  Over p = 5  Over p = 13  2 7 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} 1_{k} 1_{+} 1_{−} 0 195  1^{2} 1_{k} 3 3 1^{2} 1_{+} 5 5 1^{2} 1_{−} 13∗ 13∗ 2  2  2 2
C_{2} 4  2 2
2. 1_{k} 3_{k} 1_{+} 1_{−} 0 3 -16 1^{3} 3_{k} 3 1^{2} 1_{+} 5 5 1^{2} 1_{−} 13∗ 13∗ 2 3 3 3 2 2 2
S_{3}  3 2 2 3 3 3 3

Discriminant class: −3

∞ 3 5 13 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 13 2 7 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
3. 1_{k} 1_{i} 2_{+} 2_{+} -1 22  12 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2 2 3 2 2
S_{3} 3  2 3 2 2 2  3
4. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -65 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2 2  2 2 3
S_{3} 18  3 2  2 2 2 3 3
5. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -975 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2 2 3 2 2 3
S_{3} 15  3 2 3 2 2 2  3
6. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -195 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2  2 2 3 2 2 3
S_{3} 6  2 3 2 2 2 3 3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -2535 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2  2 2  2 2 3
S_{3} 3  3 2  2 2 2  3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -585 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2 2 3 2 2
S_{3} 3  3 2 3 2 2 2 3 3

Discriminant class: 13

∞ 3 5 13 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 13 2 7 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
9. 0_{+} 4_{+} 2_{+} 1_{+} 0 -30 -55 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  1^{2} 1_{+} 13 13 2 2 2 3 2 3 3 2
S_{3} 3  2 2  2 3 2  2