Low Degree Fields Ramifying Above {3, 11, 23}

Discriminant class: −759

∞ 3 11 23 a_{1} a_{2} a_{3}  Over p = 3  Over p = 11  Over p = 23 2 5 7 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_{i} 1_{i} 1_{k} 0 759  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{k} 23 23  2 2
C_{2} 24  2  2  2  2
2. 1_{k} 1_{i} 1_{i} 1_{k} -1 6 -3 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{k} 23 23 3 3 3 2 2  3
S_{3} 2 3 3 2 3 2 2
3. 1_{k} 5_{i} 1_{i} 1_{k} 0 -27 -153 1^{3} 5_{i} 3∗  1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{k} 23 23 3 3 3 2 2 3 3 3
S_{3} 3  2 3 3 2 3 2 3 2
4. 1_{k} 5_{i} 1_{i} 1_{k} 0 -9 -96 1^{3} 5_{i} 3∗  1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{k} 23 23  3 3 2 2 3 3
S_{3} 3  2 3  2  2 3 2
5. 1_{k} 5_{i} 1_{i} 1_{k} 0 63 141 1^{3} 5_{i} 3∗  1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{k} 23 23 3  2 2 3 3 3
S_{3} 3  2  3 2 3 2 3 2

Discriminant class: −23

∞ 3 11 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 11  Over p = 23  2 5 7 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
6. 1_{k} 4_{+} 2_{+} 1_{i} 0 33 -55 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  1^{2} 1_{i} 23∗ 23∗ 3 2 2 3 2 2 3 3
S_{3} 3  2 3 2  2 3 2 2
7. 1_{k} 4_{+} 2_{+} 1_{i} 0 33 -451 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  1^{2} 1_{i} 23∗ 23∗ 3 2 2  2 2  3
S_{3} 3  2  2 3 2 3 2 2
8. 1_{k} 4_{+} 2_{+} 1_{i} 0 99 -396 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  1^{2} 1_{i} 23∗ 23∗  2 2 3 2 2 3
S_{3} 3  2 3 2 3 2 3 2 2

Discriminant class: −3

∞ 3 11 23 a_{1} a_{2} a_{3} Over p = 3 Over p = 11 Over p = 23 2 5 7 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. 1_{k} 1_{i} 2_{+} 2_{+} 0 0 -253 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 2 3 2 3 2 3
S_{3}  2 3 2 2 2
10. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -2783 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2 3  2 3 2 3
S_{3} 3  2 3 2 2 2  3
11. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -8349 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2 3 3 2  2 3
S_{3} 6  3 2 3 2 2 2  3
12. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -17457 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2  3 2 3 2
S_{3} 6  3 2  2 2 2  3
13. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -2277 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2 3 3 2 3 2 3
S_{3} 3  3 2  2 2 2
14. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -759 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2 3  2  2
S_{3} 21  3 2 3 2 2 2

Discriminant class: 33

∞ 3 11 23 a_{1} a_{2} a_{3}  Over p = 3  Over p = 11  Over p = 23 2 5 7 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
15. 0_{+} 1_{i} 1_{k} 2_{+} -1 -38 63 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{k} 11 11 1^{3} 2_{+} ∗  3 2 2 2 3 2 3 3
S_{3}  3  2 2 2 2 2
16. 0_{+} 5_{i} 1_{k} 2_{+} 0 -207 -690 1^{3} 5_{i} 3∗  1^{2} 1_{k} 11 11 1^{3} 2_{+} ∗  2 2 2 3 2 3 3
S_{3} 3  3 2 2 2 2 2
17. 0_{+} 5_{i} 1_{k} 2_{+} 0 -207 -69 1^{3} 5_{i} 3∗ 1^{2} 1_{k} 11 11 1^{3} 2_{+} ∗ 3 2 2 2 2 3
S_{3} 3  3 3 2 2 2 2 2

Discriminant class: 253

∞ 3 11 23 a_{1} a_{2} a_{3} Over p = 3  Over p = 11  Over p = 23  2 5 7 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
18. 0_{+} 4_{+} 1_{k} 1_{i} 0 -36 -9 1^{3} 4_{+} 1  1^{2} 1_{k} 11 11 1^{2} 1_{i} 23∗ 23∗ 2 2 3 2 3 3 2 3
S_{3} 3  2 2  3 2  3 2