Quintic Fields Ramifying Above {2, 13}

Discriminant class: −26

∞ 2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  3 5 7 11 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} 9_{k} 3_{−} -1 -8 4 1^{4} 9_{k} −2∗ −∗ 1^{3} 2_{+} 1  5 5 3 4 22 4 4 32
S_{5} 13 7 1^{2} 1_{−} 13∗ 13∗ 5 5 4 22 22 4 32 2

Discriminant class: −2

∞  2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13 3 5 7 11 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. 1_{k} 11_{i} 2_{+} -2 -2 8 1^{4} 11_{i} −2 −2∗ 1^{3} 2_{+} 1  5 4 4 5 5 22 32 4
S_{5} -5 -2 2 0_{+} ∗ ∗ 4 2 22 3 4 32 5 4

Discriminant class: 2

∞  2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13 3 5 7 11 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
3. 2_{−} 11_{−} 2_{+} -2 -2 0 1^{4} 11_{−} 2 2 2^{2} 2_{+} ∗ ∗ 4 4 22 4 5 4 5 4
F_{5} -1 -2 5 4 5 4 5 4 4 4
4. 0_{+} 11_{+} 4_{+} 2 -14 -16 1^{4} 11_{+} 2 2 1^{5} 4_{+} ∗  4 4 22 4 5 4 22 4
F_{5} 41 22 5 4 22 4 5 4 4 4
5. 2_{−} 11_{−} 4_{+} 1 16 24 1^{4} 11_{−} 2 2 1^{5} 4_{+} ∗  4 4 5 4 5 4 5 4
F_{5}  70 -18  22 4 22 4 22 4 4 4
6. 2_{−} 5_{−} 4_{+} 1 3 11 1^{3} 2_{+} ∗  1^{5} 4_{+} ∗  4 32 3 4 5 32 5 2
S_{5}  -8 8  1^{2} 3_{−} 2∗ 2∗  3 4 3 4 5 32 32 4

Discriminant class: 13

∞ 2 13 a_{1} a_{2} a_{3} Over p = 2  Over p = 13 3 5 7 11 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
7. 2_{−} 4_{+} 3_{−} -1 2 -4 1^{5} 4_{+} ∗  1^{4} 3_{−} 13 13 5 4 4 4 22 4 22 5
F_{5} 1 -1 4 4 4 22 4 5 4 5

Discriminant class: 26

∞  2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  3 5 7 11 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
8. 2_{−} 11_{−} 3_{+} -2 2 8 1^{4} 11_{−} 2∗ 2∗ 1^{4} 3_{+} 13∗ 13∗ 4 22 4 5 22 5 5 4
F_{5} -7 18 4 5 4 4 4 4 5 4
9. 2_{−} 11_{−} 3_{+} -1 0 -8 1^{4} 11_{−} 2∗ 2∗ 1^{4} 3_{+} 13∗ 13∗ 4 5 4 22 22 22 5 4
F_{5} -2 -6 4 22 4 4 4 4 22 4
10. 2_{−} 11_{+} 3_{−} -1 10 6 1^{4} 11_{+} 2∗ 2∗ 1^{3} 2_{+} 1  4 5 32 3 3 5 5 4
S_{5}  -5 -3  1^{2} 1_{−} 13∗ 13∗ 32 3 4 32 4 4 22 4