Quintic Fields Ramifying Above {3, 19}

Discriminant class: −3

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

Discriminant class: 1

∞ 3 19 a_{1} a_{2} a_{3}  Over p = 3  Over p = 19 2 5 7 11 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. 2_{−} 6_{−} 2_{+} -1 1 2 1^{3} 5_{i} 3∗  2^{2} 2_{+} 1 V 5 3 5 5 3 3 3 5
A_{5}  1 5  1^{2} 1_{i} 3∗ 3∗  5 5 5 22 22 3 22 22
3. 2_{−} 4_{−} 4_{+} -2 13 -9 1^{3} 3_{i} 3∗  1^{5} 4_{+} 1  3 5 5 3 22 5 3 3
A_{5}  36 -12  1^{2} 1_{i} 3∗ 3∗  22 3 3 5 5 5 5 3
4. 2_{−} 4_{−} 4_{+} 0 0 19 1^{3} 3_{k} 3  1^{5} 4_{+} 1  5 5 22 5 3 3 5 5
A_{5}  0 -57  1^{2} 1_{k} 3 3  3 3 22 3 22 3 3 5

Discriminant class: 57

∞ 3 19 a_{1} a_{2} a_{3}  Over p = 3  Over p = 19  2 5 7 11 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
5. 2_{−} 3_{i} 3_{i} 0 1 -7 1^{3} 3_{i} 3∗  1^{4} 3_{i} 19∗ 19 5 4 3 32 4 32 4 3
S_{5}  -3 -1  2 0_{+} ∗ ∗  4 32 3 5 4 3 5 22
6. 2_{−} 5_{i} 3_{i} -1 -7 5 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  5 32 3 4 32 32 32 5
S_{5}  16 7  2 0_{+} ∗ ∗ 1^{2} 1_{i} 19∗ 19∗ 4 32 3 5 32 5 3 5
7. 2_{−} 5_{i} 3_{i} -1 -4 5 1^{3} 5_{i} 3∗  1^{4} 3_{i} 19∗ 19 5 2 5 32 4 4 4 22
S_{5}  7 -5  2 0_{+} ∗ ∗  4 4 3 5 2 3 5 5