Low Degree Fields Ramifying Above {2, 3, 11}

Discriminant class: −66

∞ 2 3 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 11 5 7 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_{i} 1_{i} 1_{k} 0 66  1^{2} 3_{i} −2 −2 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{k} 11 11  2  2 2
C_{2} 8  2  2  2
2. 1_{k} 3_{i} 3_{i} 1_{k} 0 -9 -14 1^{2} 3_{i} −2 −2 1^{3} 3_{i} 3∗  1^{2} 1_{k} 11 11 3  3 3 2 3 2 2
S_{3}  2 3 2  3 2 3 3
3. 1_{k} 3_{i} 5_{i} 1_{k} 0 18 -48 1^{2} 3_{i} −2 −2 1^{3} 5_{i} 3∗  1^{2} 1_{k} 11 11 3 3 3 3 2  2 2
S_{3}  2 3 2  3 2 3
4. 1_{k} 3_{i} 5_{i} 1_{k} 0 -9 -30 1^{2} 3_{i} −2 −2 1^{3} 5_{i} 3∗  1^{2} 1_{k} 11 11  3 3 3 2 3 2 2
S_{3}  2  2  2 3 3
5. 1_{k} 3_{i} 5_{i} 1_{k} 0 -27 -78 1^{2} 3_{i} −2 −2 1^{3} 5_{i} 3∗  1^{2} 1_{k} 11 11 3 3  2 3 2 2
S_{3}  2 3 2  3 2  3

Discriminant class: −33

∞ 2 3 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 11  5 7 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} 2_{i} 1_{k} 1_{i} 0 33  1^{2} 2_{i} −1 −1 1^{2} 1_{k} 3 3 1^{2} 1_{i} 11∗ 11∗ 2  2  2
C_{2} 4  2  2 2
7. 1_{k} 2_{i} 3_{k} 1_{i} 0 6 -12 1^{2} 2_{i} −1 −1 1^{3} 3_{k} 3  1^{2} 1_{i} 11∗ 11∗ 2 3 2 3  3 3 2
S_{3}  3 3 3 3 2 3 2 2

Discriminant class: −22

∞ 2 3 11 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 11 5 7 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} 3_{−} 4_{+} 1_{k} 0 6 -32 1^{2} 3_{−} 2∗ 2∗ 1^{3} 4_{+} ∗ 1^{2} 1_{k} 11 11 2 2 2 3 3 3 3
S_{3}  2 2 3 3 2 2 3 2

Discriminant class: −11

∞ 2 3 11 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 11  5 7 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
9. 1_{k} 2_{+} 4_{+} 1_{i} 0 6 -10 1^{3} 2_{+} ∗  1^{3} 4_{+} 1  1^{2} 1_{i} 11∗ 11∗  2 2 2 2 3 2 3
S_{3} 3  3 2 2 3 3 3 2
10. 1_{k} 2_{+} 4_{+} 1_{i} 0 -12 -28 1^{3} 2_{+} ∗  1^{3} 4_{+} 1  1^{2} 1_{i} 11∗ 11∗ 3 2 2 2 2 3 2
S_{3} 3  2 2 3 3  2 3

Discriminant class: −3

∞ 2 3 11 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 11 5 7 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
11. 1_{k} 2_{+} 1_{i} 2_{+} -1 -7 13 1^{3} 2_{+} ∗  1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗  2 3  2 3 2 2 3
S_{3}  3 2  2 2 2 3 3
12. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -22 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  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 -132 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2  3 2  2 2
S_{3} 3  3 2 3 2 2 2 3 3
14. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -396 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 2 3 3 2 3 2 2 3
S_{3} 3  2 3 2 2 2 3 3
15. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -198 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 2 3 3 2 2 2 3
S_{3} 6  3 2 3 2 2 2 3
16. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -66 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 2 3 2 3 2 2
S_{3} 6  2 3 2 2 2 3

Discriminant class: −1

∞ 2 3 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 11 5 7 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. 1_{k} 2_{i} 4_{+} 2_{+} 0 33 -22 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗  1^{3} 2_{+} ∗  3 2  3 2 2  2
S_{3} 6  3 2 2 3 2 3 2
18. 1_{k} 2_{i} 4_{+} 2_{+} 0 33 -176 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗ 1^{3} 2_{+} ∗ 2 3 2 2 3 2
S_{3} 3  3 3 2 2 3 2 3 2

Discriminant class: 2

∞ 2 3 11 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 11 5 7 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_{+} 3_{+} 4_{+} 2_{+} 0 -66 -176 1^{2} 3_{+} 2 2 1^{3} 4_{+} ∗  1^{3} 2_{+} ∗  2 3 2 3 2 3 2 3
S_{3}  2  2  2 2 2 2

Discriminant class: 6

∞ 2 3 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 11 5 7 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
20. 0_{+} 3_{k} 3_{i} 2_{+} 0 -33 -66 1^{2} 3_{k} −2∗ −2∗ 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  3 2 2 2 3  3 2
S_{3}  2 2 3  2 2
21. 0_{+} 3_{k} 5_{i} 2_{+} 0 -99 -330 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 2 2 3 3  2
S_{3}  2 2 3 3 3 2 2 3
22. 0_{+} 3_{k} 5_{i} 2_{+} 0 -99 66 1^{2} 3_{k} −2∗ −2∗ 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 3 2 2 2 3 3 2
S_{3}  2 2 3 3 3 2 2 3

Discriminant class: 66

∞ 2 3 11 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 11  5 7 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
23. 0_{+} 3_{+} 1_{k} 1_{i} 0 -66  1^{2} 3_{+} 2 2 1^{2} 1_{k} 3 3 1^{2} 1_{i} 11∗ 11∗  2  2 2
C_{2} 2  2  2  2