Low Degree Fields Ramifying Above {3, 5, 19}

Discriminant class: −95

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

Discriminant class: −15

∞ 3 5 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 19 2 7 11 13 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
2. 1_{k} 1_{k} 1_{−} 2_{+} -1 -6 45 1^{2} 1_{k} 3 3 1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} 1  3 2 2 2 3  2 3
S_{3} 3  2 2 2 3 3 2  2
3. 1_{k} 3_{k} 1_{−} 2_{+} 0 57 -38 1^{3} 3_{k} 3  1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} 1  2 2 2 3 3 2
S_{3} 3  2 2 2 3 3 2 3 2
4. 1_{k} 3_{k} 1_{−} 2_{+} 0 -57 -209 1^{3} 3_{k} 3  1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} 1  3 2 2 2  3 2 3
S_{3} 3  2 2 2 3  2 3 2

Discriminant class: −3

∞ 3 5 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 19 2 7 11 13 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} 3_{i} 2_{+} 2_{+} 0 0 -475 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2  2 3 2 2 2 3
S_{3} 3  2  2 2 2 3 3
6. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -95 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2 3 2 2 2
S_{3} 3  3 2 3 2 2 2  3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -285 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2  2 2 2 3
S_{3} 15  2  2 2 2  3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1425 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2  2 2 2
S_{3} 18  3 2 3 2 2 2 3 3
9. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -4275 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 3 2 3 2 2 2 3
S_{3} 3  3 2 3 2 2 2 3 3
10. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -855 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2  2 3 2 2 2 3
S_{3} 27  3 2 3 2 2 2  3

Discriminant class: 5

∞ 3 5 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 19 2 7 11 13 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_{+} 4_{+} 1_{+} 2_{+} 0 -57 -76 1^{3} 4_{+} ∗  1^{2} 1_{+} 5 5 1^{3} 2_{+} 1  2 2  2 2 2 3 3
S_{3} 3  2 3 2 2 2  3 2

Discriminant class: 57

∞ 3 5 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 5  Over p = 19  2 7 11 13 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
12. 0_{+} 1_{k} 2_{+} 1_{i} -1 -8 -3 1^{2} 1_{k} 3 3 1^{3} 2_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 3 3 2 2 2 2 3 2
S_{3}  2 3  2 3 3 3 2

Discriminant class: 285

∞ 3 5 19 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5  Over p = 19  2 7 11 13 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
13. 0_{+} 1_{i} 1_{−} 1_{i} 0 -285  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{i} 19∗ 19∗ 2 2 2  2
C_{2} 2  2  2
14. 0_{+} 5_{i} 1_{−} 1_{i} 0 -18 -3 1^{3} 5_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{i} 19∗ 19∗ 2 2 2 3 3 3 3 2
S_{3}  3 2  2 3 3 3