Low Degree Fields Ramifying Above {3, 13, 19}

Discriminant class: −3

∞ 3 13 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 13 Over p = 19 2 5 7 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} 3_{i} 2_{+} 2_{+} 0 0 -247 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 3 2 2 2 2
S_{3} 9  3 2  2 2 2  3
2. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -3211 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 3 2 2 2 2 3
S_{3} 9  2 3 2 2 2 3
3. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -14079 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} 1 2 2 3 2 2 2 2 3
S_{3} 18  3 2 3 2 2 2 3
4. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -741 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2  2 2 2 2 3
S_{3} 72  2 3 2 2 2  3
5. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -2223 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2 3 2 2 2 2 3
S_{3} 9  3 2 3 2 2 2  3
6. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -9633 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} 1  2 2  2 2 2 2
S_{3} 18  3 2  2 2 2 3

Discriminant class: 1

∞ 3 13 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 13 Over p = 19 2 5 7 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. 0_{+} 4_{+} 2_{+} 2_{+} 0 -741 -2717 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3  3 3 3
C_{3} 9  3  3 3 3 3  3
8. 0_{+} 4_{+} 2_{+} 2_{+} 0 -741 6175 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 -741 7163 1^{3} 4_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3  3 3 3  3
C_{3} 9  3 3 3 3 3  3 3
10. 0_{+} 4_{+} 2_{+} 2_{+} 0 -741 4940 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

Discriminant class: 741

∞ 3 13 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 13  Over p = 19 2 5 7 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_{+} 1_{k} 1_{−} 1_{k} 0 -741  1^{2} 1_{k} 3 3 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19 2  2  2 2
C_{2} 2  2  2
12. 0_{+} 3_{k} 1_{−} 1_{k} 0 -12 -3 1^{3} 3_{k} 3  1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19 2 3 2 3 2 2 3
S_{3}  3 2 3  3 2 3 3