Low Degree Fields Ramifying Above {3, 19, 23}

Discriminant class: −1311

∞ 3 19 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 19 Over p = 23 2 5 7 11 13 17 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_{k} 1_{k} 0 1311  1^{2} 1_{k} 3 3 1^{2} 1_{k} 19 19 1^{2} 1_{k} 23 23  2  2  2
C_{2} 28  2 2 2  2
2. 1_{k} 3_{k} 1_{k} 1_{k} 0  -3 -21 1^{3} 3_{k} 3  1^{2} 1_{k} 19 19 1^{2} 1_{k} 23 23 3 3 2 3 2 3 3 2
S_{3}  3 3 2 2 2  2 3

Discriminant class: −3

∞ 3 19 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 19 Over p = 23 2 5 7 11 13 17 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} 3_{i} 2_{+} 2_{+} 0 0 -437 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 3 2 2 2 3
S_{3} 6  3 2 3 2 2 2 3 3
4. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -8303 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 3 2 2 3
S_{3} 6  3 2  2 2 2 3
5. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -24909 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2  2 2
S_{3} 54  2 3 2 2 2 3
6. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1311 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 2 2 3
S_{3} 9  3 2  2 2 2 3 3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -30153 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 2 2 3
S_{3} 3  3 2 3 2 2 2 3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -3933 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 2 3 2 2
S_{3} 3  2 3 2 2 2 3 3

Discriminant class: 57

∞ 3 19 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 19  Over p = 23 2 5 7 11 13 17 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_{+} 3_{k} 1_{i} 2_{+} 0 -69 -92 1^{3} 3_{k} 3  1^{2} 1_{i} 19∗ 19∗ 1^{3} 2_{+} ∗  2 3 2 2 2  2
S_{3}  2 3 3 2  3 3 2