Low Degree Fields Ramifying Above {2, 3, 23}

Discriminant class: −138

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23  5 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} 3_{k} 1_{i} 1_{i} 0 138  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{i} 23∗ 23∗ 2  2  2  2
C_{2} 8  2 2  2 2  2
2. 1_{k} 3_{k} 3_{i} 1_{i} 0 15 -34 1^{2} 3_{k} −2∗ −2∗ 1^{3} 3_{i} 3∗  1^{2} 1_{i} 23∗ 23∗ 2 3 3 2  2 3 2
S_{3}  2 2 3 2 2 3 2
3. 1_{k} 3_{k} 5_{i} 1_{i} 0 63 -66 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗  1^{2} 1_{i} 23∗ 23∗ 2 3  2  2 3 2
S_{3}  3 2 2  2 2 3 2
4. 1_{k} 3_{k} 5_{i} 1_{i} 0 45 -114 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗  1^{2} 1_{i} 23∗ 23∗ 2 3 3 2  2  2
S_{3} 4  3 2 2 3 2 2 3 2
5. 1_{k} 3_{k} 5_{i} 1_{i} 0 -9 -42 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗  1^{2} 1_{i} 23∗ 23∗ 2  3 2  2 3 2
S_{3}  3 2 2 3 2 2  2

Discriminant class: −69

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23  5 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
6. 1_{k} 2_{i} 1_{k} 1_{i} 0 69  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{k} 3 3 1^{2} 1_{i} 23∗ 23∗  2  2 2
C_{2} 8  2 2  2
7. 1_{k} 2_{i} 3_{k} 1_{i} 0 -6 -20 1^{2} 2_{i} −∗ −∗ 1^{3} 3_{k} 3  1^{2} 1_{i} 23∗ 23∗  3 2 3 3 3 2 2
S_{3}  2 2 3 3 3 3 2

Discriminant class: −46

∞ 2 3 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 23  5 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
8. 1_{k} 3_{+} 4_{+} 1_{i} 0 -3 -94 1^{2} 3_{+} 2 2 1^{3} 4_{+} ∗  1^{2} 1_{i} 23∗ 23∗ 3 2 3 2 2  2 3
S_{3}  3 3  3 2 3 3

Discriminant class: −6

∞ 2 3 23 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 23 5 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
9. 1_{k} 3_{−} 1_{k} 2_{+} -1 8 40 1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 3 3 1^{3} 2_{+} ∗  3 3 2 2 2 3 3
S_{3}  2 2 2 2 3 3 2 2
10. 1_{k} 3_{−} 3_{k} 2_{+} 0 69 -322 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  3  2 2 2 3 3
S_{3} 3  2 2 2 2  3 2 2
11. 1_{k} 3_{−} 3_{k} 2_{+} 0 69 -138 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  3 3 3 2 2 2 3
S_{3} 3  2 2 2 2 3 3 2 2

Discriminant class: −3

∞ 2 3 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 23 5 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. 1_{k} 2_{+} 1_{i} 2_{+} 0 0 -46 1^{3} 2_{+} ∗ 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗ 2 3 2 3 2 2
S_{3}  3 2 3 2 2 2 3 3
13. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -92 1^{3} 2_{+} ∗ 1^{3} 3_{i} 3∗ 1^{3} 2_{+} ∗ 2 2 2 3 2
S_{3} 3  3 2 3 2 2 2 3
14. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -828 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 3 2 3 2  2 3
S_{3} 6  2  2 2 2 3
15. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -138 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 3 2  2 3 2 3
S_{3} 3  2 3 2 2 2 3 3
16. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -414 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2  2 3 2 3 2 3
S_{3} 6  3 2  2 2 2 3 3
17. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -276 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 3 2 3 2 3 2 3
S_{3} 3  3 2 3 2 2 2 3

Discriminant class: −2

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23 5 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
18. 1_{k} 3_{i} 4_{+} 2_{+} 0 69 -46 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 3 2 2
S_{3} 6  2 3  2 2 3 2 3
19. 1_{k} 3_{i} 4_{+} 2_{+} 0 69 -874 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  2 2 3 2  2 2
S_{3} 3  2 3 3 2 2  2

Discriminant class: −1

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23 5 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
20. 1_{k} 2_{i} 4_{+} 2_{+} 0 69 -230 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗  1^{3} 2_{+} ∗  2 2 3 3 2  2
S_{3} 6  3 3 2 2 3 2  2
21. 1_{k} 2_{i} 4_{+} 2_{+} 0 69 -598 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗  1^{3} 2_{+} ∗  3 2 2  3 2 3 2
S_{3} 3  3 3 2 2 3 2 3 2

Discriminant class: 69

∞ 2 3 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 23 5 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
22. 0_{+} 2_{+} 5_{i} 1_{k} 0 -36 -78 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{2} 1_{k} 23 23 3 2 3  3 2 2 3
S_{3}  2 2 2 2  2 2 2
23. 0_{+} 2_{+} 5_{i} 1_{k} 0 -18 -6 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{2} 1_{k} 23 23 3 2  3  2 2 3
S_{3}  2 2 2 2 3 2 2 2
24. 0_{+} 2_{+} 5_{i} 1_{k} 0 -36 60 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 1^{2} 1_{k} 23 23 2 3 3 3 2 2
S_{3}  2 2 2 2 3 2 2 2

Discriminant class: 138

∞ 2 3 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 23 5 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
25. 0_{+} 3_{−} 1_{k} 1_{k} 0 -138  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 3 3 1^{2} 1_{k} 23 23 2 2 2 2
C_{2} 2  2  2 2