Low Degree Fields Ramifying Above {3, 7, 23}

Discriminant class: −483

∞ 3 7 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 23 2 5 11 13 17 19 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} 0 483  1^{2} 1_{k} 3 3 1^{2} 1_{k} 7 7 1^{2} 1_{k} 23 23 2 2  2 2  2 2
C_{2} 4  2  2  2
2. 1_{k} 3_{k} 1_{k} 1_{k} 0 15 12 1^{3} 3_{k} 3  1^{2} 1_{k} 7 7 1^{2} 1_{k} 23 23 2 2 3 2 2 3 2 2
S_{3} 2 3 2 3 3 2

Discriminant class: −3

∞ 3 7 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 23 2 5 11 13 17 19 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} 1_{i} 2_{+} 2_{+} 0 0 -161 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 2  2 3 2 3
S_{3} 3  3 2  2 2 2 3
4. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -1127 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 2 3 2 3 2 3
S_{3} 3  3 2 3 2 2 2 3 3
5. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -3381 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 2  2  2
S_{3} 6  2  2 2 2 3 3
6. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -10143 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 2 3 2 3 2 3
S_{3} 9  3 2 3 2 2 2 3 3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1449 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 2 3 2 3 2
S_{3} 3  2 3 2 2 2 3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -483 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 2 3 2 2 3
S_{3} 12  3 2 3 2 2 2 3

Discriminant class: 21

∞ 3 7 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 23 2 5 11 13 17 19 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_{+} 1_{k} 1_{i} 2_{+} -1 -15 -6 1^{2} 1_{k} 3 3 1^{2} 1_{i} 7∗ 7∗ 1^{3} 2_{+} ∗  2 3 2 2 3 2 2 2
S_{3}  3 3 3 3 2 3 2 3