Defining polynomial
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$( x^{5} + x^{2} + 1 )^{4} + \left(4 x^{3} + 4 x^{2} + 6 x + 4\right) ( x^{5} + x^{2} + 1 )^{3} + \left(2 x^{4} + 2 x^{3} + 2 x^{2} + 2 x + 2\right) ( x^{5} + x^{2} + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x^{2} + 4 x\right) ( x^{5} + x^{2} + 1 ) + 12 x^{2} + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $20$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $40$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, 3]$ |
| Visible Swan slopes: | $[1,2]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}\rangle$ |
| Rams: | $(1, 3)$ |
| Jump set: | $[1, 3, 7]$ |
| Roots of unity: | $62 = (2^{ 5 } - 1) \cdot 2$ |
Intermediate fields
| 2.5.1.0a1.1, 2.5.2.10a7.1 |
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
\( x^{5} + x^{2} + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{4} + \left(4 t + 4\right) x^{3} + \left(2 t^{3} + 2 t^{2} + 2 t + 2\right) x^{2} + \left(4 t^{3} + 4 t^{2} + 4 t\right) x + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + (t^3 + t^2 + t + 1)$,$(t^3 + t^2 + t + 1) z + (t^4 + t^3 + t^2)$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $2560$ |
| Galois group: | $C_2^8:C_{10}$ (as 20T256) |
| 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 |