Quintic Fields Unramified Outside of {∞, 5, 11}

Discriminant class: −55

∞ 5 11 a_{1} a_{2} a_{3} Over p = 5  Over p = 11 2 3 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} 9_{−} 1_{k}  0  0 -25 1^{5} 9_{−} 5  1^{2} 1_{k} 11 11 5 32 3 22 5 4 4 2
S_{5}  0 55  2 0_{+} ∗ ∗ 3 2 32 5 32 4 22 32
2. 1_{k} 9_{−} 3_{k}  0 -25 -50 1^{5} 9_{−} 5  1^{3} 2_{+} ∗  5 4 5 5 5 32 2 2
S_{5}  350 -335  1^{2} 1_{k} 11 11 5 4 4 22 32 4 3 2

Discriminant class: −11

∞ 5 11 a_{1} a_{2} a_{3} Over p = 5  Over p = 11  2 3 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
3. 1_{k} 6_{+} 3_{i}  0 -10 -20 1^{5} 6_{+} 1  1^{4} 3_{i} 11∗ 11 32 5 2 4 4 32 5 4
S_{5}  -15 7  3 3 32 4 3 5 3 32

Discriminant class: 1

∞ 5 11 a_{1}  a_{2} a_{3} Over p = 5  Over p = 11  2 3 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. 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
5. 0_{+} 8_{+} 0_{+}  0 -10 -5 1^{5} 8_{+} 1  5 0_{+} 1  5 5  5 5 5 5 5
C_{5}  10  -1  5 5 5  5 5 5 5
6. 0_{+} 8_{+} 4_{+}  0 -110 605 1^{5} 8_{+} 1  1^{5} 4_{+} 1  5 5 5 5 5 5 5 5
C_{5} 5  -990 451  5  5 5 5 5
7. 0_{+} 8_{+} 4_{+}  0 -110 220 1^{5} 8_{+} 1  1^{5} 4_{+} 1  5 5 5 5  5
C_{5} 5  1485 -1276  5 5 5  5 5 5 5
8. 0_{+} 8_{+} 4_{+}  0 -110 55 1^{5} 8_{+} 1  1^{5} 4_{+} 1  5 5 5 5  5 5 5
C_{5} 5  660 -649  5  5  5  5
9. 0_{+} 8_{+} 4_{+}  0 -110 -495 1^{5} 8_{+} 1  1^{5} 4_{+} 1  5  5  5 5 5 5
C_{5} 5  -440 -99  5 5 5  5 5
10. 2_{−} 6_{+} 2_{−}  0  5 -5 1^{5} 6_{+} 1  1^{2} 1_{i} 11∗ 11∗ 5 22 5  5 22 22 22
D_{5}  5  3  1^{2} 1_{i} 11∗ 11∗ 5 22 22 5 22 22 5 22
11. 2_{−} 8_{+} 2_{−}  0 -15 -15 1^{5} 8_{+} 1  1^{2} 1_{i} 11∗ 11∗ 22 5 22 22 22 22 5 22
D_{5} 5  110 143  1^{2} 1_{i} 11∗ 11∗ 5  22 22  5 5 22
12. 0_{+} 6_{+} 4_{+}  0 -55 55 1^{5} 6_{+} 1  1^{5} 4_{+} 1  22 22 22 22 22 5 22 5
D_{5} 5  110  33  5 22 5 22 22 22 5 5

Discriminant class: 5

∞ 5 11 a_{1}  a_{2} a_{3} Over p = 5  Over p = 11  2 3 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
13. 2_{−} 7_{−} 0_{+} 0  5 0 1^{5} 7_{−} 5  5 0_{+} 1  4 4 4 4 4 22 4 22
F_{5}  5  -1  5 4 5 4 4 4 22 5

Discriminant class: 5 (continued)

∞ 5 11 a_{1}  a_{2} a_{3} Over p = 5  Over p = 11  2 3 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
14. 2_{−} 3_{−} 4_{+} -2  6 2 1^{4} 3_{−} 5 5 1^{5} 4_{+} 1  4 4 4 4 4 22 4 22
F_{5} 5  4  1  5 4 5 4 4 4 22 5
15. 2_{−} 9_{−} 0_{+} 0  0 0 1^{5} 9_{−} 5  5 0_{+} 1  4 4 4 4 4 22 4 22
F_{5}  0  -5  4 5 4 4 4 22 5
16. 2_{−} 5_{−} 4_{+} 0  0 0 1^{5} 5_{−} 5  1^{5} 4_{+} 1  4 4 4 4 4 22 4 22
F_{5} 5  0 -11  5 4 5 4 4 4 22
17. 0_{+} 9_{−} 2_{−} 0 -25 0 1^{5} 9_{−} 5  1^{2} 1_{i} 11∗ 11∗ 4 4 4 4 4 5 4 5
F_{5}  125  75  1^{2} 1_{i} 11∗ 11∗ 5 4 22 4 4 4 22 22
18. 2_{−} 7_{−} 4_{+} 0 -55 0 1^{5} 7_{−} 5  1^{5} 4_{+} 1  4 4 4 4 4 22 4 22
F_{5} 10  605 -803  5 4 5 4 4 4 22 5
19. 2_{−} 7_{−} 4_{+} 0 -55 0 1^{5} 7_{−} 5  1^{5} 4_{+} 1  4 4 4 4 4 22 4 22
F_{5} 10  605 -847  5 4 5 4 4 4 22
20. 2_{−} 7_{−} 4_{+} 0  55 0 1^{5} 7_{−} 5  1^{5} 4_{+} 1  4 4 4 4 4 22 4 22
F_{5} 5  605 429  4 5 4 4 4 22 5
21. 2_{−} 7_{−} 4_{+} 0  55 0 1^{5} 7_{−} 5  1^{5} 4_{+} 1  4 4 4 4 4 22 4 22
F_{5} 5  605 -121  5 4  4 4 4 22 5
22. 2_{−} 9_{−} 4_{+} 0  0 0 1^{5} 9_{−} 5  1^{5} 4_{+} 1  4 4 4 4 4 22 4 22
F_{5} 5  0 -605  5 4 5 4 4 4 22 5
23. 2_{−} 9_{−} 4_{+} 0  0 0 1^{5} 9_{−} 5  1^{5} 4_{+} 1  4 4 4 4 4 22 4 22
F_{5} 25  0 -275  5 4 5 4 4 4 22 5
24. 2_{−} 9_{−} 4_{+} 0  0 0 1^{5} 9_{−} 5  1^{5} 4_{+} 1  4 4 4 4 4 22 4 22
F_{5} 15  0 -55  5 4  4 4 4 22 5
25. 2_{−} 9_{−} 4_{+} 0  0 0 1^{5} 9_{−} 5  1^{5} 4_{+} 1  4 4 4 4 4 22 4 22
F_{5} 40  0 -6875  5 4 5 4 4 4 22 5
26. 2_{−} 5_{−} 2_{+} 0  0 -5 1^{5} 5_{−} 5  1^{3} 2_{+} ∗  32 32 4 4 2 5 32 5
S_{5}  5  -5  2 0_{+} ∗  ∗ 5 4 22 2 4 4 3 3
27. 2_{−} 9_{−} 4_{+} 0  0 0 1^{5} 9_{−} 5  1^{5} 4_{+} 1  32 2 2 32 32 5 4 5
S_{5} 5  275 -165  5 4 5 4 4 32 5 22