Low Degree Fields Ramifying Above {3, 71}

Discriminant class: −71

∞ 3 71 a_{1} a_{2} a_{3} Over p = 3  Over p = 71  2 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d  s 29 31 37 41 43 47 53 59
1. 1_{k} 4_{+} 1_{i} 0 -9 -45 1^{3} 4_{+} 1  1^{2} 1_{i} 71∗ 71∗ 3  2 2 2 2 3 2
S_{3} 3  3 2  2 3 2 2 2

Discriminant class: −3

∞ 3 71 a_{1} a_{2} a_{3} Over p = 3 Over p = 71 2 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 41 43 47 53 59
2. 1_{k} 1_{i} 2_{+} 0 0 -71 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗ 2 2 2 3 2 3 2
S_{3}  2 3 3 2 3 2 2 2
3. 1_{k} 5_{i} 2_{+} 0 0 -213 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 2 3 2  2 3 2
S_{3} 21  2  3 2  2 2 2
4. 1_{k} 5_{i} 2_{+} 0 0 -639 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 2 3 2 3 2  2
S_{3} 18  2 3  2 3 2 2 2

Discriminant class: 213

∞ 3 71 a_{1} a_{2} a_{3} Over p = 3 Over p = 71 2 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 41 43 47 53 59
5. 0_{+} 1_{i} 1_{k} 0 -213 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{k} 71 71 2 2 2 2
C_{2}  2 2
6. 0_{+} 5_{i} 1_{k} 0 -18 -15 1^{3} 5_{i} 3∗  1^{2} 1_{k} 71 71 2 2 2 3 2 3 3
S_{3}  2 2 3  3 3 3