Quintic Fields Ramifying Above {7, 11}

Discriminant class: 1

∞ 7 11 a_{1} a_{2} a_{3}  Over p = 7  Over p = 11 2 3 5 13 17 19 23 29
G h  a_{4} a_{5}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 2_{−} 2_{−} 4_{+} -2 6 -9 1^{2} 1_{i} 7∗ 7∗ 1^{5} 4_{+} 1  5 22 22 22 22 22 5 5
D_{5} 5 4 1 1^{2} 1_{i} 7∗ 7∗ 22 5 22 5 22 5 22 22

Discriminant class: 77

∞ 7 11 a_{1} a_{2} a_{3} Over p = 7  Over p = 11 2 3 5 13 17 19 23 29
G h  a_{4} a_{5}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
2. 2_{−} 3_{k} 3_{k} -2 2 -3 1^{3} 2_{+} 1  1^{3} 2_{+} ∗  4 32 4 5 5 3 5 32
S_{5}  7 -4  1^{2} 1_{k} 7 7 1^{2} 1_{k} 11 11 4 5 3 4 2 5 4 5