Quintic Fields Ramifying Above {2, 3}

Discriminant class: −6

∞  2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3 5 7 11 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_{+} 5_{i} 0 -2 -4 1^{4} 11_{+} 2∗ 2∗ 1^{3} 5_{i} 3∗  5 5 3 32 32 32 32 3
S_{5} -9 -4 2 0_{+} ∗ ∗ 5 4 4 4 4 22 5 4

Discriminant class: 2

∞  2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3 5 7 11 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_{−} 11_{−} 4_{+} -2 -2 8 1^{4} 11_{−} 2 2 1^{5} 4_{+} ∗  4 5 4 4 5 4 5 4
F_{5} -1 -10 22 4 22 4 22 4 4 4
3. 2_{−} 9_{−} 6_{+} -1 -2 6 1^{4} 9_{−} 2 −1 1^{3} 5_{i} 3∗  4 5 4 32 5 4 5 4
S_{5} -6 6 1^{2} 1_{k} 3 3 3 32 5 32 5 4 32 32
4. 2_{−} 11_{−} 6_{+} -1 4 -12 1^{4} 11_{−} 2 −2 1^{3} 5_{i} 3∗  32 3 32 4 5 32 5 4
S_{5} 12 -12 1^{2} 1_{k} 3 3 3 32 3 32 22 2 4 2

Discriminant class: 3

∞  2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3 5 7 11 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_{−} 10_{i} 5_{i} -2 2 4 1^{4} 10_{i} −∗ −2∗ 1^{3} 5_{i} 3∗  32 4 5 3 32 4 5 4
S_{5} -5 2 2 0_{+} ∗ ∗ 32 5 4 32 3 32 22 3

Discriminant class: 6

∞  2 3 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  5 7 11 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
6. 2_{−} 11_{i} 5_{i} -2 4 0 1^{4} 11_{i} −2∗ 2∗ 1^{3} 4_{+} 1  5 4 4 32 4 5 3 5
S_{5}  -6 12  1^{2} 1_{i} 3∗ 3∗ 4 32 32 3 5 3 32 2