Low Degree Fields Ramifying Above {3, 7, 11}

Discriminant class: −231

∞ 3 7 11 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 11 2 5 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} 1_{k} 1_{k} 1_{k} 0 231  1^{2} 1_{k} 3 3 1^{2} 1_{k} 7 7 1^{2} 1_{k} 11 11  2  2  2
C_{2} 12  2 2  2
2. 1_{k} 1_{k} 1_{k} 1_{k} -1 0 3 1^{2} 1_{k} 3 3 1^{2} 1_{k} 7 7 1^{2} 1_{k} 11 11 3 3 3 2 3 2 3 2
S_{3} 3 2 2 3 2 3 3

Discriminant class: −3

∞ 3 7 11 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 11 2 5 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
3. 1_{k} 1_{i} 2_{+} 2_{+} -1 26 -20 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 3 2 2 2 3
S_{3} 3  3 2 3 2 2 2 3
4. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -77 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 2 2 2
S_{3} 3  3 2 3 2 2 2 3 3
5. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -693 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 1^{3} 2_{+} ∗ 2 2 3 2 3 2 2 3
S_{3} 3  2 3 2 2 2 3 3
6. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -231 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 2 2 3
S_{3} 18  3 2  2 2 2 3 3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -2541 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 2 2
S_{3} 3  3 2  2 2 2 3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1617 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2  2 3 2 2 3
S_{3} 6  2 3 2 2 2 3

Discriminant class: 21

∞ 3 7 11 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 11 2 5 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. 0_{+} 3_{k} 1_{i} 2_{+} 0 -33 -44 1^{3} 3_{k} 3  1^{2} 1_{i} 7∗ 7∗ 1^{3} 2_{+} ∗  2 3 2  2 2 2 2
S_{3}  3 3  3 2 3 2 3

Discriminant class: 77

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