Low Degree Fields Ramifying Above {2, 3, 5}

Discriminant class: −30

∞ 2 3 5 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 5 7 11 13 17 19 23 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_{+} 1_{i} 1_{+} 0 30  1^{2} 3_{+} 2 2 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 5 5 2  2
C_{2} 4  2  2  2
2. 1_{k} 3_{+} 3_{i} 1_{+} 0 3 -6 1^{2} 3_{+} 2 2 1^{3} 3_{i} 3∗  1^{2} 1_{+} 5 5 2 3 3 3 2 3 3 3
S_{3}  2 3 3 2  2
3. 1_{k} 3_{+} 5_{i} 1_{+} 0 27 -18 1^{2} 3_{+} 2 2 1^{3} 5_{i} 3∗  1^{2} 1_{+} 5 5 2  3 3 2  3 3
S_{3}  3 2 3 3 2 3 2 3
4. 1_{k} 3_{+} 5_{i} 1_{+} 0 -18 -48 1^{2} 3_{+} 2 2 1^{3} 5_{i} 3∗  1^{2} 1_{+} 5 5 2 3  3 2 3 3
S_{3}  3 2 3  2 3 2 3
5. 1_{k} 3_{+} 5_{i} 1_{+} 0 18 -24 1^{2} 3_{+} 2 2 1^{3} 5_{i} 3∗  1^{2} 1_{+} 5 5 2 3 3  2 3  3
S_{3}  3 2  3 2 3 2 3

Discriminant class: −10

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  7 11 13 17 19 23 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} 3_{k} 4_{+} 1_{−} 0 -3 -22 1^{2} 3_{k} −2∗ −2∗ 1^{3} 4_{+} ∗  1^{2} 1_{−} 5∗ 5∗ 3  3 2 3 3 2 2
S_{3}  3 2 3 3 3 2 2

Discriminant class: −5

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 23 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
7. 1_{k} 2_{i} 4_{+} 1_{+} 0 -3 -8 1^{2} 2_{i} −∗ −∗ 1^{3} 4_{+} 1  1^{2} 1_{+} 5 5 3 2 2 2 2 3 3 2
S_{3} 3 2 3 3 2 2 3 3

Discriminant class: −3

∞ 2 3 5 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 5 7 11 13 17 19 23 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} 2_{+} 1_{i} 2_{+} -1 -3 -3 1^{3} 2_{+} ∗ 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗ 3 2 3 2 3 2 2 3
S_{3}  2 3 2 2 2 3 3
9. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -20 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  2 3 2  2 2 3
S_{3} 3  3 2 3 2 2 2  3
10. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -60 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  3 2  2 3 2 2
S_{3} 3  2 3 2 2 2  3
11. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -30 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  3 2 3 2  2 2
S_{3} 3  3 2 3 2 2 2 3 3
12. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -150 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  3 2 3 2 3 2 2 3
S_{3} 3  3 2  2 2 2  3
13. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -90 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2  2 3 2 2 3
S_{3} 3  3 2  2 2 2 3 3

Discriminant class: −2

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 23 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
14. 1_{k} 3_{i} 4_{+} 2_{+} 0 -30 -80 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 3 2 3
15. 1_{k} 3_{i} 4_{+} 2_{+} 0 15 -10 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  2 3 2 3  2 2 2
S_{3} 3  2  3 2 2 3 2

Discriminant class: 2

∞ 2 3 5 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 23 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
16. 0_{+} 3_{+} 4_{+} 2_{+} 0 -30 -40 1^{2} 3_{+} 2 2 1^{3} 4_{+} ∗ 1^{3} 2_{+} ∗ 3 2 2 3 2 2
S_{3}  2 3 2 3 2 2 2 2

Discriminant class: 3

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 23 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
17. 0_{+} 2_{i} 3_{k} 2_{+} 0 -15 -20 1^{2} 2_{i} −∗ −∗ 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  2 3 3 2 2 3 2 2
S_{3}  3 2 2 3 2 3 3 2

Discriminant class: 5

∞ 2 3 5 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 23 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_{+} 2_{+} 4_{+} 1_{+} 0 -12 -14 1^{3} 2_{+} ∗  1^{3} 4_{+} ∗  1^{2} 1_{+} 5 5 2 3 2 2 3 2 3 3
S_{3}  2 3 2 2 2 3  2

Discriminant class: 15

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5  7 11 13 17 19 23 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
19. 0_{+} 2_{i} 1_{i} 1_{−} 0 -15  1^{2} 2_{i} −1 −1 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{−} 5∗ 5∗  2  2 2 2 2
C_{2} 2  2 2  2
20. 0_{+} 2_{i} 5_{i} 1_{−} 0 -18 -12 1^{2} 2_{i} −1 −1 1^{3} 5_{i} 3∗  1^{2} 1_{−} 5∗ 5∗ 3 3 2 3 2 2 2 2
S_{3}  2 2 3 2 3  3 3

Discriminant class: 30

∞ 2 3 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 5 7 11 13 17 19 23 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
21. 0_{+} 3_{i} 1_{k} 1_{+} 0 -30  1^{2} 3_{i} −2 −2 1^{2} 1_{k} 3 3 1^{2} 1_{+} 5 5  2  2  2
C_{2} 2  2 2 2 2 2 2 2