Defining polynomial
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$( x^{5} + x^{2} + 1 )^{4} + \left(2 x^{2} + 2\right) ( x^{5} + x^{2} + 1 ) + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $20$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $20$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_1$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{4}{3}, \frac{4}{3}]$ |
| Visible Swan slopes: | $[\frac{1}{3},\frac{1}{3}]$ |
| Means: | $\langle\frac{1}{6}, \frac{1}{4}\rangle$ |
| Rams: | $(\frac{1}{3}, \frac{1}{3})$ |
| Jump set: | $[1, 2, 5]$ |
| 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: |
\( x^{4} + \left(2 t + 2\right) x + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + (t^4 + t^3 + t)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $7680$ |
| Galois group: | $C_2^8.(C_5\times S_3)$ (as 20T374) |
| Inertia group: | not computed |
| Wild inertia group: | not computed |
| Galois unramified degree: | not computed |
| Galois tame degree: | not computed |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: | not computed |