Defining polynomial
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$( x^{5} + x^{2} + 1 )^{4} + \left(4 x^{3} + 4 x + 4\right) ( x^{5} + x^{2} + 1 )^{3} + 2 ( x^{5} + x^{2} + 1 )^{2} + 4 x ( x^{5} + x^{2} + 1 ) + 8 x^{2} + 2$
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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 $c$: | $40$ |
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| Discriminant root field: | $\Q_{2}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{2})$: | $C_2$ | |
| Visible Artin slopes: | $[2, 3]$ | |
| Visible Swan slopes: | $[1,2]$ | |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}\rangle$ | |
| Rams: | $(1, 3)$ | |
| Jump set: | $[1, 5, 9]$ | |
| Roots of unity: | $124 = (2^{ 5 } - 1) \cdot 2^{ 2 }$ |
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Intermediate fields
| $\Q_{2}(\sqrt{-1})$, 2.5.1.0a1.1, 2.5.2.10a1.1 |
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: |
\( x^{4} + 4 t x^{3} + 2 x^{2} + 4 t^{4} x + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + (t^3 + t^2 + 1)$,$(t^3 + t^2 + 1) z + (t^4 + t^3)$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 2, 0]$ |
Invariants of the Galois closure
| Galois degree: | not computed |
| Galois group: | not computed |
| 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 |