Low Degree Fields Ramifying Above {3, 7}

Discriminant class: −7

∞ 3 7 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 2 5 11 13 17 19 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 1_{k} 4_{+} 1_{i} 0 -3 -5 1^{3} 4_{+} ∗  1^{2} 1_{i} 7∗ 7∗ 3 2 3 2 2 2
S_{3}  2 3 2 3 2 3 2 2

Discriminant class: −3

∞ 3 7 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 2 5 11 13 17 19 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
2. 1_{k} 3_{i} 2_{+} 0 0 -7 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 2 2 2 3 2 2 2
S_{3} 3  3 3 2 3 2 2 2 3
3. 1_{k} 5_{i} 2_{+} 0 0 -21 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 2  2 3 2 2
S_{3} 3  3 3 2  2 2 2 3
4. 1_{k} 5_{i} 2_{+} 0 0 -63 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 2 3 2 3 2 2
S_{3} 6  2 3 2 2 2 3

Discriminant class: 1

∞ 3 7 a_{1} a_{2} a_{3} Over p = 3  Over p = 7 2 5 11 13 17 19 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
5. 0_{+} 4_{+} 2_{+} 0 -21 35 1^{3} 4_{+} 1 1^{3} 2_{+} 1 3 3 3 3 3
C_{3} 3  3 3 3 3 3 3 3 3
6. 0_{+} 4_{+} 2_{+} 0 -21 -28 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3 3 3
C_{3} 3  3 3 3  3

Discriminant class: 21

∞ 3 7 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 2 5 11 13 17 19 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
7. 0_{+} 1_{k} 1_{i} 0 -21  1^{2} 1_{k} 3 3 1^{2} 1_{i} 7∗ 7∗ 2  2 2  2 2 2
C_{2}  2  2  2