Defining polynomial
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$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 50 x^{2}\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$
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Invariants
| Base field: | $\Q_{5}$ |
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| Degree $d$: | $20$ |
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| Ramification index $e$: | $5$ |
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| Residue field degree $f$: | $4$ |
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| Discriminant exponent $c$: | $36$ |
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| Discriminant root field: | $\Q_{5}(\sqrt{2})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{5})$: | $C_1$ | |
| Visible Artin slopes: | $[\frac{9}{4}]$ | |
| Visible Swan slopes: | $[\frac{5}{4}]$ | |
| Means: | $\langle1\rangle$ | |
| Rams: | $(\frac{5}{4})$ | |
| Jump set: | undefined | |
| Roots of unity: | $624 = (5^{ 4 } - 1)$ |
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Intermediate fields
| $\Q_{5}(\sqrt{2})$, 5.4.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 5.4.1.0a1.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{4} + 4 x^{2} + 4 x + 2 \)
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|
| Relative Eisenstein polynomial: |
\( x^{5} + \left(50 t^{3} + 100\right) x + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + (t^3 + 3 t^2 + 3 t + 2)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[5, 0]$ |
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 |