Low Degree Fields Ramifying Above {3, 5, 23}

Discriminant class: −115

∞ 3 5 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 23 2 7 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} 4_{+} 1_{−} 1_{k} 0 -18 -63 1^{3} 4_{+} ∗ 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{k} 23 23 2 2 2 3 2 3 3
S_{3}  3 3 3 2 3 3 2 3

Discriminant class: −23

∞ 3 5 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 5  Over p = 23  2 7 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
2. 1_{k} 4_{+} 2_{+} 1_{i} 0 15 -35 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  1^{2} 1_{i} 23∗ 23∗ 3 2 2 3 2 2
S_{3} 3  2 3 2  2  2 2
3. 1_{k} 4_{+} 2_{+} 1_{i} 0 45 -45 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  1^{2} 1_{i} 23∗ 23∗ 3 2 2  2 2 3 3
S_{3} 6  2 3 2 3 2  2 2
4. 1_{k} 4_{+} 2_{+} 1_{i} 0 15 -80 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  1^{2} 1_{i} 23∗ 23∗  2 2 3 2 2 3 3
S_{3} 3  2  2 3 2  2 2

Discriminant class: −3

∞ 3 5 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 23 2 7 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
5. 1_{k} 1_{i} 2_{+} 2_{+} -1 -38 132 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2  2 3 2 3 2 3
S_{3}  3 2  2 2 2 3 3
6. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -115 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 3 2 3 2  2 3
S_{3} 3  3 2 3 2 2 2 3 3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -345 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 3 2 3 2 3 2
S_{3} 3  3 2  2 2 2 3 3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -5175 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 3 2  2  2
S_{3} 18  3 2 3 2 2 2 3 3
9. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1035 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 2 2 3 2 3
S_{3} 6  2 3 2 2 2 3 3
10. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1725 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 3 2 3 2 3 2 3
S_{3} 18  2 3 2 2 2 3 3

Discriminant class: 5

∞ 3 5 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 23 2 7 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
11. 0_{+} 4_{+} 1_{+} 2_{+} 0 -138 23 1^{3} 4_{+} ∗  1^{2} 1_{+} 5 5 1^{3} 2_{+} ∗  2 2 3 2 2  3
S_{3}  2  2 2 2 3 3 2

Discriminant class: 345

∞ 3 5 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 5  Over p = 23  2 7 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
12. 0_{+} 1_{k} 1_{+} 1_{i} 0 -345  1^{2} 1_{k} 3 3 1^{2} 1_{+} 5 5 1^{2} 1_{i} 23∗ 23∗  2 2 2 2
C_{2} 2  2  2 2 2