Defining polynomial
|
\(x^{10} - 20 x^{9} - 350 x^{8} - 190 x^{5} - 4100 x^{4} - 10975\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 5 }) }$: | $5$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| $\Q_{5}(\sqrt{2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{2} + 4 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{5} + \left(15 t + 20\right) x^{4} + 100 t + 105 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{4} + 3t + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_5\times D_5$ (as 10T6) |
| Inertia group: | Intransitive group isomorphic to $C_5^2$ |
| Wild inertia group: | $C_5^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2]$ |
| Galois mean slope: | $48/25$ |
| Galois splitting model: | $x^{10} + 110 x^{8} - 825 x^{7} + 5280 x^{6} - 84260 x^{5} + 296450 x^{4} - 3000800 x^{3} + 18227440 x^{2} - 37758050 x + 396143473$ |