Low Degree Fields Ramifying Above {3, 19, 29}

Discriminant class: −551

∞ 3 19 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 19  Over p = 29  2 5 7 11 13 17 23 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 -21 -55 1^{3} 4_{+} 1  1^{2} 1_{k} 19 19 1^{2} 1_{−} 29∗ 29∗ 3  2 2 2 3 3
S_{3} 6  3 3 2 2 2 2 2 2

Discriminant class: −19

∞ 3 19 29 a_{1} a_{2}  a_{3} Over p = 3  Over p = 19  Over p = 29 2 5 7 11 13 17 23 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} 4_{+} 1_{i} 2_{+} 0 -261 -2088 1^{3} 4_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 1^{3} 2_{+} ∗  2 3 3 3 2 3 3 2
S_{3} 3  2 2  2 2 3 2
3. 1_{k} 4_{+} 1_{i} 2_{+} 0 -174 -1769 1^{3} 4_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 1^{3} 2_{+} ∗  2 3  3 2 3 3 2
S_{3} 30  2 2 3 3 2 2  2

Discriminant class: −3

∞ 3 19 29 a_{1} a_{2} a_{3} Over p = 3 Over p = 19 Over p = 29 2 5 7 11 13 17 23 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
4. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -551 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2  2 2 3
S_{3} 3  3 2  2 2 2 3
5. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -10469 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 2 2 3
S_{3} 24  3 2 3 2 2 2  3
6. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1653 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 3 2 2 3
S_{3} 18  3 2 3 2 2 2 3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -31407 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2  2 2
S_{3} 9  2 3 2 2 2  3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -47937 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 3 2 2 3
S_{3} 3  3 2  2 2 2  3
9. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -4959 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 2 2
S_{3} 3  2 3 2 2 2 3

Discriminant class: 1653

∞ 3 19 29 a_{1} a_{2} a_{3} Over p = 3 Over p = 19 Over p = 29 2 5 7 11 13 17 23 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
10. 0_{+} 1_{i} 1_{k} 1_{+} 0 -1653  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{k} 19 19 1^{2} 1_{+} 29 29 2 2  2  2
C_{2} 2  2 2  2 2
11. 0_{+} 5_{i} 1_{k} 1_{+} 0 -90 -321 1^{3} 5_{i} 3∗  1^{2} 1_{k} 19 19 1^{2} 1_{+} 29 29 2 2 3 3 2  2 3
S_{3} 2  3 2 2  3 3 2 2