Low Degree Fields Ramifying Above {3, 5, 11}

Discriminant class: −55

∞ 3 5 11 a_{1} a_{2} a_{3} Over p = 3  Over p = 5  Over p = 11  2 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} 4_{+} 1_{+} 1_{i} 0 -3 -13 1^{3} 4_{+} ∗  1^{2} 1_{+} 5 5 1^{2} 1_{i} 11∗ 11∗ 3 3  3 2 2 2
S_{3}  2 2 3 2 2 3 2 2

Discriminant class: −15

∞ 3 5 11 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 11 2 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
2. 1_{k} 1_{k} 1_{−} 2_{+} -1 4 -9 1^{2} 1_{k} 3 3 1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} ∗ 3 2 2 3 3 2 3
S_{3}  2 2 2 3 3 2 3 2
3. 1_{k} 3_{k} 1_{−} 2_{+} 0 -33 -88 1^{3} 3_{k} 3  1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} ∗  2 2 3 3 3 2 3
S_{3} 3  2 2 2 3  2 3 2
4. 1_{k} 3_{k} 1_{−} 2_{+} 0 33 -11 1^{3} 3_{k} 3  1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} ∗  3 2 2  3  2
S_{3} 3  2 2 2 3 3 2  2

Discriminant class: −3

∞ 3 5 11 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 11 2 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
5. 1_{k} 1_{i} 2_{+} 2_{+} 0 0 -55 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 3 2 3 2 2 3
S_{3}  3 2 3 2 2 2 3 3
6. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -275 1^{3} 3_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 3 3 2 3 2 2
S_{3} 3  3 2 3 2 2 2 3 3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -165 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 3 3 2 3 2 2 3
S_{3} 9  3 2 3 2 2 2 3 3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1815 1^{3} 5_{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
9. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -495 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 3  2  2 2
S_{3} 9  2  2 2 2 3 3
10. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -825 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2  3 2  2 2 3
S_{3} 3  2  2 2 2 3 3

Discriminant class: 33

∞ 3 5 11 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 11 2 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. 0_{+} 3_{i} 2_{+} 1_{k} 0 -15 -15 1^{3} 3_{i} 3∗ 1^{3} 2_{+} ∗ 1^{2} 1_{k} 11 11 3 2 2 3 2 2 3
S_{3}  3 2 2 2 2 2 3
12. 0_{+} 5_{i} 2_{+} 1_{k} 0 -45 -60 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{2} 1_{k} 11 11 2 2 3 2 2 3 3
S_{3}  3 3 2 2 2 2 2 3
13. 0_{+} 5_{i} 2_{+} 1_{k} 0 -45 -105 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{2} 1_{k} 11 11 3 2 2  2 2 3 3
S_{3}  3  2 2 2 2 2 3

Discriminant class: 165

∞ 3 5 11 a_{1} a_{2} a_{3} Over p = 3  Over p = 5  Over p = 11 2 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
14. 0_{+} 1_{k} 1_{−} 1_{k} 0 -165  1^{2} 1_{k} 3 3 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{k} 11 11 2  2 2
C_{2} 2  2  2 2 2