Quintic Fields Unramified Outside of {∞, 2, 11}

Discriminant class: −22

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

Discriminant class: 1

∞ 2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11 3 5 7 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. 0_{+} 0_{+} 4_{+} -1 -4 3 5 0_{+} 1  1^{5} 4_{+} 1  5 5 5 5 5 5  5
C_{5}  3 -1  5 5 5  5 5 5 5
3. 2_{−} 6_{−} 4_{+} -1 -4 -8 1^{2} 3_{i} −2 −2 1^{5} 4_{+} 1  5 22 22 22 5 5 22 22
D_{5} 5 3 -23 1^{2} 3_{i} −2 −2 22 22 5 5 22 22 5 22

Discriminant class: 2

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11 3 5 7 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
4. 2_{−} 11_{+} 2_{−} 0 2 -4 1^{4} 11_{+} 2 2 2^{2} 2_{−} ∗ ∗ 4 4 5 4 22 4 22 4
F_{5} -1 -4 5 4 5 4 5 4 4 4

Discriminant class: 22

∞  2 11 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  3 5 7 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
5. 2_{−} 11_{i} 3_{i} -2 0 -8 1^{4} 11_{i} −2∗ −2 1^{4} 3_{i} 11∗ 11 5 32 5 3 4 32 2 3
S_{5}  2 -4  32 4 2 4 32 4 3 5