Quintic Fields Ramifying Above {5, 13}

Discriminant class: 1

∞ 5 13 a_{1} a_{2} a_{3} Over p = 5  Over p = 13 2 3 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. 2_{−} 6_{+} 2_{−} 0 5 -10 1^{5} 6_{+} 1  2^{2} 2_{−} 1 V 5 3 3 5 3 3 3 5
A_{5} 3 0 -45 5 22 3 5 5 5 3 22

Discriminant class: 5

∞ 5 13 a_{1}  a_{2} a_{3} Over p = 5  Over p = 13 2 3 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. 2_{−} 5_{−} 2_{+}  0  5  0 1^{5} 5_{−} 5  2^{2} 2_{+} ∗ ∗ 4 4 4 22 4 5 4 22
F_{5}  5  -3  22 4 22 4 4 4  5
3. 2_{−} 7_{−} 2_{+}  0  5  0 1^{5} 7_{−} 5  2^{2} 2_{+} ∗ ∗ 4 4 4 22 4 5 4 22
F_{5}  5 -16  22 4 22 4 4 4 5 5
4. 2_{−} 5_{−} 4_{+}  0  0  0 1^{5} 5_{−} 5  1^{5} 4_{+} ∗  4 4 4 5 4 22 4 22
F_{5} 4  0 -13  5 4 5 4 4 4 22
5. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  4 4 4 5 4 22 4 22
F_{5}  0 -845  5 4 5 4 4 4 22 5
6. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  4 4 4 5 4 22 4 22
F_{5} 2  0 -325  5 4  4 4 4 22 5
7. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  4 4 4 5 4 22 4 22
F_{5} 2  0 -8125  5 4 5 4 4 4 22 5
8. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  4 4 4  4 22 4 22
F_{5} 2  0 -65  5 4 5 4 4 4 22 5
9. 2_{−} 7_{−} 2_{+}  0  0 -5 1^{5} 7_{−} 5  2^{2} 2_{+} ∗ ∗ 32 32 2 5 2 5 4 5
S_{5}  15  -7  3 32 5 4 4 32 3 3
10. 2_{−} 7_{−} 2_{+}  0  0 -15 1^{5} 7_{−} 5  1^{3} 2_{+} 1  4 4 2 5 32 22 4 3
S_{5}  15  -4  2 0_{+} ∗ ∗ 3 32 22 32 32 4 5 22
11. 2_{−} 9_{−} 2_{+}  0  0 -50 1^{5} 9_{−} 5  1^{3} 2_{+} 1  4 2 4 5 32 3 4 3
S_{5}  -150 -315  2 0_{+} ∗ ∗ 22 4 5 4 32 4 5 3
12. 2_{−} 7_{−} 4_{+}  0  0 -130 1^{5} 7_{−} 5  1^{5} 4_{+} ∗  2 4 32 3 32 5 32 3
S_{5} 5  -585 -1222  3 4 3 32 2 4 5 3
13. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  32 32 32 5 4 5 4 3
S_{5} 2  325 -65  22 4 3 2 32 32 5 5

Discriminant class: 13

∞ 5 13 a_{1} a_{2} a_{3} Over p = 5  Over p = 13 2 3 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
14. 2_{−} 6_{+} 3_{−} 0 10 -35 1^{5} 6_{+} ∗  1^{4} 3_{−} 13 13 4 22 4 4 5 4 5 5
F_{5} -15 72 4 4 4 4 22 4 5
15. 2_{−} 6_{+} 3_{−} 0 10 -35 1^{5} 6_{+} ∗  1^{4} 3_{−} 13 13 4 22 4 4 5 4 5 5
F_{5}  -15 -266  4 4 4 5 4 22 4
16. 2_{−} 8_{+} 3_{−} 0 40 -45 1^{5} 8_{+} ∗  1^{4} 3_{−} 13 13 4  4 4 22 4 22 5
F_{5}  85 36  4 4 4 22 4 5 4 5

Discriminant class: 13 (continued)

∞ 5 13 a_{1} a_{2} a_{3} Over p = 5  Over p = 13 2 3 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
17. 2_{−} 8_{+} 3_{−} 0 10 -35 1^{5} 8_{+} ∗  1^{4} 3_{−} 13 13 4 5 4 4 22 4 22
F_{5} -15 -58 4 4 4 22 4 5 4 5
18. 2_{−} 6_{+} 3_{−} 0 -10 -20 1^{5} 6_{+} ∗  1^{4} 3_{−} 13 13 32 5 4 4 3 2 3 22
S_{5}  115 -121  2 2 32 5 32 22 4 3

Discriminant class: 65

∞ 5 13 a_{1} a_{2} a_{3}  Over p = 5  Over p = 13  2 3 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
19. 2_{−} 5_{+} 3_{−} 0  5 -15 1^{5} 5_{+} 5∗  1^{3} 2_{+} 1  5 32 22 2 4 32 4 3
S_{5}  -15 -49  1^{2} 1_{−} 13∗ 13∗ 4 22 4 32 3 32 4 3
20. 2_{−} 7_{+} 3_{−} 0  5 50 1^{5} 7_{+} 5∗  1^{3} 2_{+} 1  5 4 5 4 32 4 4 5
S_{5}  50 -205  1^{2} 1_{−} 13∗ 13∗ 4 5 4 2 5 32 4 5
21. 2_{−} 7_{+} 3_{−} 0  5 -15 1^{5} 7_{+} 5∗  1^{3} 2_{+} 1  3 4 5 4 32 4 4 5
S_{5}  50 -88  1^{2} 1_{−} 13∗ 13∗ 2 5 4 32 3 4 4 22