Low Degree Fields Ramifying Above {3, 17, 19}

Discriminant class: −323

∞ 3 17 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 17  Over p = 19  2 5 7 11 13 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 -12 -35 1^{3} 4_{+} 1  1^{2} 1_{+} 17 17 1^{2} 1_{i} 19∗ 19∗ 2 2 2 2 2 2 3 3
S_{3} 3 3 3 3 2 2 2 2

Discriminant class: −51

∞ 3 17 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 17  Over p = 19 2 5 7 11 13 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 -25 64 1^{2} 1_{k} 3 3 1^{2} 1_{−} 17∗ 17∗ 1^{3} 2_{+} 1  2 3 2 3  3  2
S_{3} 3  2 3 3 2 2 2 2 3
3. 1_{k} 3_{k} 1_{−} 2_{+} 0 -57 -228 1^{3} 3_{k} 3  1^{2} 1_{−} 17∗ 17∗ 1^{3} 2_{+} 1  2 3 2 3 3 3 3 2
S_{3} 3  2 3  2 2 2 2 3
4. 1_{k} 3_{k} 1_{−} 2_{+} 0 -114 -475 1^{3} 3_{k} 3  1^{2} 1_{−} 17∗ 17∗ 1^{3} 2_{+} 1  2  2 3 3  3 2
S_{3} 6  2  3 2 2 2 2

Discriminant class: −3

∞ 3 17 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 17 Over p = 19 2 5 7 11 13 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} 1_{i} 2_{+} 2_{+} 0 0 -323 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2  2 3 2 2 3
S_{3} 3  2  2 2 2 3 3
6. 1_{k} 1_{i} 2_{+} 2_{+} -1 108  -12 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 3 2  2 2
S_{3} 3  3 2 3 2 2 2
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -18411 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 2 2 3 2 2 3
S_{3} 27  3 2 3 2 2 2
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -16473 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 3 2 3 2 2 3
S_{3} 72  2  2 2 2
9. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -2907 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 3
10. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -969 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 2 3 2 3 2 2
S_{3} 27  3 2 3 2 2 2 3 3

Discriminant class: 17

∞ 3 17 19 a_{1} a_{2} a_{3} Over p = 3 Over p = 17 Over p = 19 2 5 7 11 13 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 -95 1^{3} 4_{+} ∗  1^{2} 1_{+} 17 17 1^{3} 2_{+} 1  3 2 2 2  2 2 2
S_{3} 3  2 2 3  3 3 2

Discriminant class: 57

∞ 3 17 19 a_{1} a_{2} a_{3} Over p = 3  Over p = 17  Over p = 19  2 5 7 11 13 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_{+} 3_{k} 2_{+} 1_{i} 0 -51 -119 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 3 2  2 2 2  2
S_{3}  2 3 3 2  3 3 2

Discriminant class: 969

∞ 3 17 19 a_{1} a_{2} a_{3}  Over p = 3  Over p = 17  Over p = 19  2 5 7 11 13 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 -969  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{−} 17∗ 17∗ 1^{2} 1_{i} 19∗ 19∗  2  2  2
C_{2} 2  2  2  2 2
14. 0_{+} 5_{i} 1_{−} 1_{i} 0 -27 -3 1^{3} 5_{i} 3∗  1^{2} 1_{−} 17∗ 17∗ 1^{2} 1_{i} 19∗ 19∗ 3 3 2 3 2  2 3
S_{3}  3 2  2 3 3 2 2