Low Degree Fields Ramifying Above {3, 5, 7}

Discriminant class: −35

∞ 3 5 7 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 7 2 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} 4_{+} 1_{−} 1_{k} 0 -12 -19 1^{3} 4_{+} 1 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{k} 7 7 2 3 3 3 2 2 2
S_{3} 3  2 2 2 3 2 2 2 2

Discriminant class: −7

∞ 3 5 7 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 7 2 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
2. 1_{k} 4_{+} 2_{+} 1_{i} 0 15 -5 1^{3} 4_{+} ∗  1^{3} 2_{+} ∗  1^{2} 1_{i} 7∗ 7∗ 3  2 2 2 3 3 2
S_{3} 3  3 2  2 3 2 2
3. 1_{k} 4_{+} 2_{+} 1_{i} 0 15 -40 1^{3} 4_{+} ∗  1^{3} 2_{+} ∗  1^{2} 1_{i} 7∗ 7∗  3 2 2 2 3 3 2
S_{3} 3  2 3 2 3 2 2 3

Discriminant class: −3

∞ 3 5 7 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 7 2 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
4. 1_{k} 1_{i} 2_{+} 2_{+} 0 0 -35 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 3 2 3 2 2
S_{3} 3  3 2  2 2 2 3 3
5. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -175 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 3 2 3 2 2 3
S_{3} 3  2 3 2 2 2  3
6. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -315 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 3 2  2 2 3
S_{3} 3  3 2 3 2 2 2 3 3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -105 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 2 2 3 2 2 3
S_{3} 6  2 3 2 2 2 3 3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -735 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 3 2 3 2 2 3
S_{3} 3  3 2  2 2 2  3
9. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -525 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2  2  2 2
S_{3} 9  3 2 3 2 2 2  3

Discriminant class: 105

∞ 3 5 7 a_{1} a_{2} a_{3} Over p = 3 Over p = 5 Over p = 7 2 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
10. 0_{+} 1_{i} 1_{+} 1_{k} 0 -105  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 5 5 1^{2} 1_{k} 7 7  2  2 2  2 2
C_{2} 2  2  2 2  2 2
11. 0_{+} 5_{i} 1_{+} 1_{k} 0 -27 -9 1^{3} 5_{i} 3∗  1^{2} 1_{+} 5 5 1^{2} 1_{k} 7 7 3 2 3 2 2 3 2 2
S_{3}  2  2 2  3 2 2