Quintic Fields Unramified Outside of {∞, 11, 19}

Discriminant class: −19

∞ 11 19 a_{1} a_{2} a_{3} Over p = 11  Over p = 19  2 3 5 7 13 17 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. 1_{k} 4_{+} 3_{i} -1 -4 -8 1^{5} 4_{+} 1  1^{3} 2_{+} 1  4 4 5 22 4 3 5 4
S_{5} 5  -19 -12  1^{2} 1_{i} 19∗ 19∗ 32 2 32 3 5 32 4 5

Discriminant class: 1

∞ 11 19 a_{1} a_{2} a_{3}  Over p = 11  Over p = 19  2 3 5 7 13 17 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. 0_{+} 4_{+} 0_{+} -1 -4 3 1^{5} 4_{+} 1  5 0_{+} 1  5 5 5 5 5 5  5
C_{5}  3 -1  5 5 5  5 5 5 5
3. 2_{−} 4_{+} 2_{−} -2 -5 2 1^{5} 4_{+} 1  1^{2} 1_{i} 19∗ 19∗ 22 22 5 5 22 5 5 22
D_{5} 5 48 -32 1^{2} 1_{i} 19∗ 19∗ 22 22 22 5 5 22 22 5
4. 2_{−} 2_{−} 4_{+} -1 8 -13 1^{2} 1_{i} 11∗ 11∗ 1^{5} 4_{+} 1  22 5 5 22 22 22 5 22
D_{5}  7 2  1^{2} 1_{i} 11∗ 11∗  5 5 22 22 5 5 5 22