Defining polynomial
|
\(x^{10} + 10 x^{9} + 74 x^{8} + 320 x^{7} + 1104 x^{6} + 2752 x^{5} + 6176 x^{4} + 12096 x^{3} + 17712 x^{2} + 15968 x + 8416\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| 2.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.5.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{5} + x^{2} + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + \left(2 t^{4} + 2 t + 2\right) x + 4 t^{4} + 4 t^{2} + 6 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + t^{4} + t + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^4:C_5$ (as 10T8) |
| Inertia group: | Intransitive group isomorphic to $C_2^4$ |
| Wild inertia group: | $C_2^4$ |
| Unramified degree: | $5$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2, 2]$ |
| Galois mean slope: | $15/8$ |
| Galois splitting model: | $x^{10} - x^{8} - 4 x^{6} + 3 x^{4} + 3 x^{2} - 1$ |