Quintic Fields Unramified Outside of {∞, 2, 19}

Discriminant class: −38

∞  2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19 3 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} 11_{−} 3_{k} 0 -14 -4 1^{4} 11_{−} 2∗ −2∗ 1^{3} 2_{+} 1  22 4 3 4 5 5 5 3
S_{5} 51 36 1^{2} 1_{k} 19 19 32 5 4 32 5 5 5 32

Discriminant class: 1

∞ 2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19 3 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_{−} 4_{−} 4_{+} 2 -6 -10 1^{2} 2_{i} −1 −1 1^{5} 4_{+} 1  22 5 22 22 5 5 22 5
D_{5}  17 12  1^{2} 2_{i} −1 −1  22 5 5 22 22 5 22 5
3. 2_{−} 6_{−} 4_{+} -1 8 -32 1^{4} 6_{−} 1  1^{5} 4_{+} 1  5 22 3 5 3 22 5 5
A_{5} 3  45 -17  5 5 3 5 3 5 5 3

Discriminant class: 2

∞  2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19 3 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
4. 2_{−} 11_{+} 2_{−} -1 6 -6 1^{4} 11_{+} 2 2 2^{2} 2_{−} ∗ ∗ 4 4 22 4 4 5 22 4
F_{5} 11 -3 22 4 5 4 5 4 4 4

Discriminant class: 19

∞ 2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19 3 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_{−} 4_{k} 3_{k} -2 3 -6 2^{2} 4_{k} −∗ ∗ 1^{3} 2_{+} 1  5 5 4 32 4 3 4 2
S_{5}  3 2  1^{2} 1_{k} 19 19 3 4 4 32 4 32 3 22