Defining polynomial
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$( x^{5} + x^{2} + 1 )^{4} + 8 ( x^{5} + x^{2} + 1 )^{3} + 4 ( x^{5} + x^{2} + 1 )^{2} + 26$
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Invariants
| Base field: | $\Q_{2}$ |
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| Degree $d$: | $20$ |
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| Ramification index $e$: | $4$ |
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| Residue field degree $f$: | $5$ |
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| Discriminant exponent $c$: | $55$ |
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| Discriminant root field: | $\Q_{2}(\sqrt{2\cdot 5})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{2})$ $=$ $\Gal(K/\Q_{2})$: | $C_{20}$ | |
| This field is Galois and abelian over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[3, 4]$ | |
| Visible Swan slopes: | $[2,3]$ | |
| Means: | $\langle1, 2\rangle$ | |
| Rams: | $(2, 4)$ | |
| Jump set: | $[1, 3, 7]$ | |
| Roots of unity: | $62 = (2^{ 5 } - 1) \cdot 2$ |
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Intermediate fields
| $\Q_{2}(\sqrt{2\cdot 5})$, 2.1.4.11a1.18, 2.5.1.0a1.1, 2.5.2.15a1.10 |
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: |
\( x^{4} + 8 x^{3} + 4 x^{2} + 26 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + (t^4 + t^3 + t + 1)$,$(t^4 + t^3 + t + 1) z + t$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[8, 4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $20$ |
| Galois group: | $C_{20}$ (as 20T1) |
| 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 |