Defining polynomial
|
$( x^{5} + x^{2} + 1 )^{2} + 2 x ( x^{5} + x^{2} + 1 ) + 2$
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $10$ |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2]$ |
| Visible Swan slopes: | $[1]$ |
| Means: | $\langle\frac{1}{2}\rangle$ |
| Rams: | $(1)$ |
| Jump set: | $[1, 3]$ |
| Roots of unity: | $62 = (2^{ 5 } - 1) \cdot 2$ |
Intermediate fields
| 2.5.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.5.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{5} + x^{2} + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + \left(2 t^{4} + 2 t^{3} + 2 t + 2\right) x + 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + (t^2 + 1)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $80$ |
| Galois group: | $C_2^4:C_5$ (as 10T8) |
| Inertia group: | Intransitive group isomorphic to $C_2^4$ |
| Wild inertia group: | $C_2^4$ |
| Galois unramified degree: | $5$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 2, 2]$ |
| Galois Swan slopes: | $[1,1,1,1]$ |
| Galois mean slope: | $1.875$ |
| Galois splitting model: | $x^{10} - 4 x^{8} + 2 x^{6} + 5 x^{4} - 2 x^{2} - 1$ |