Defining polynomial
|
\(x^{10} + 5 x^{7} + 20 x^{6} + 5\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $10$ |
| Ramification index $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{5})$: | $C_1$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible Artin slopes: | $[\frac{7}{4}]$ |
| Visible Swan slopes: | $[\frac{3}{4}]$ |
| Means: | $\langle\frac{3}{5}\rangle$ |
| Rams: | $(\frac{3}{2})$ |
| Jump set: | undefined |
| Roots of unity: | $4 = (5 - 1)$ |
Intermediate fields
| $\Q_{5}(\sqrt{5})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{5}$ |
| Relative Eisenstein polynomial: |
\( x^{10} + 5 x^{7} + 20 x^{6} + 5 \)
|
Ramification polygon
| Residual polynomials: | $z^5 + 2$,$2 z^2 + 1$ |
| Associated inertia: | $1$,$2$ |
| Indices of inseparability: | $[6, 0]$ |
Invariants of the Galois closure
| Galois degree: | $200$ |
| Galois group: | $D_5:F_5$ (as 10T17) |
| Inertia group: | $C_5:F_5$ (as 10T10) |
| Wild inertia group: | $C_5^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{5}{4}, \frac{7}{4}]$ |
| Galois Swan slopes: | $[\frac{1}{4},\frac{3}{4}]$ |
| Galois mean slope: | $1.63$ |
| Galois splitting model: | $x^{10} - 120 x^{7} - 220 x^{6} - 96 x^{5} + 5600 x^{4} + 4200 x^{3} + 31185 x^{2} + 3360 x + 3584$ |