Quintic Fields Unramified Outside of {∞, 5, 7}

Discriminant class: −35

∞ 5 7 a_{1} a_{2} a_{3}  Over p = 5  Over p = 7 2 3 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} 7_{+} 3_{i}  0 -10 20 1^{5} 7_{+} 5∗  1^{4} 3_{i} 7 7∗ 32 3 22 3 5 4 4 22
S_{5}  -15 -171  4 32 32 2 22 32 32 32

Discriminant class: 1

∞ 5 7 a_{1} a_{2} a_{3} Over p = 5  Over p = 7  2 3 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. 0_{+} 8_{+} 0_{+} 0 -10 -5 1^{5} 8_{+} 1  5 5 5 5 5 5 5 5
C_{5} 10 -1 5 5 5 5 5 5 5
3. 2_{−} 6_{+} 2_{−} 0 0 -5 1^{5} 6_{+} 1  1^{2} 1_{k} 7 7 22 5 5 5 5 22 22 5
D_{5} 10 -4 1^{2} 1_{k} 7 7 22 22 22 22 22 22 22
4. 2_{−} 8_{+} 2_{−} 0 15 -5 1^{5} 8_{+} 1  1^{2} 1_{i} 7∗ 7∗ 5 22 5 22 22 22 5
D_{5} 35 -91 1^{2} 1_{i} 7∗ 7∗ 22 22 5 22 5 22 22

Discriminant class: 5

∞ 5 7  a_{1}  a_{2} a_{3} Over p = 5  Over p = 7  2 3 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_{−} 7_{−} 0_{+}  0  5  0 1^{5} 7_{−} 5  4 0_{+} ∗ ∗ 4 4 5 4 4 22 4 22
F_{5}  5  -1  5 4 5 4 4 4 22 5
6. 2_{−} 3_{−} 4_{+}  0  0  0 1^{4} 3_{−} 5 5 1^{5} 4_{+} ∗  4 4 5 4 4 22 4 22
F_{5}  0  -7  5 4 5 4 4 4 22 5
7. 2_{−} 9_{−} 0_{+}  0  0  0 1^{5} 9_{−} 5  4 0_{+} ∗ ∗ 4 4 5 4 4 22 4 22
F_{5}  0  -5  4 5 4 4 4 22 5
8. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  4 4 5 4 4 22 4 22
F_{5} 2  0 -245  5 4  4 4 4 22
9. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  4 4 5 4 4 22 4 22
F_{5}  0 -35  5 4 5 4 4 4 22 5
10. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  4 4  4 4 22 4 22
F_{5}  0 -175  5 4 5 4 4 4 22 5
11. 2_{−} 9_{−} 4_{+}  0  0  0 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  4 4 5 4 4 22 4 22
F_{5}  0 -4375  5 4 5 4 4 4 22 5
12. 2_{−} 7_{−} 2_{+}  0  0 -10 1^{5} 7_{−} 5  1^{3} 2_{+} 1  32 4 22 32 4 5 32 3
S_{5}  15  29  2 0_{+} ∗ ∗ 5 4 22 32 2 4 22 5
13. 2_{−} 9_{−} 2_{+}  0  0 -100 1^{5} 9_{−} 5  1^{2} 1_{k} 7 7 4 32 3 32 4 5 4 5
S_{5}  -100 -55  1^{2} 1_{i} 7∗ 7∗ 3 2 3 4 32 2 5 22
14. 2_{−} 9_{−} 4_{+}  0  0 -175 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  2 4 5 2 4 3 4 3
S_{5} 5  -1050 -3640  3 4 5 2 32 32 22 5