Defining polynomial
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$( x^{2} + 4 x + 2 )^{10} + \left(10 x + 20\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$
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Invariants
| Base field: | $\Q_{5}$ |
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| Degree $d$: | $20$ |
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| Ramification index $e$: | $10$ |
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| Residue field degree $f$: | $2$ |
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| Discriminant exponent $c$: | $26$ |
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| Discriminant root field: | $\Q_{5}(\sqrt{2})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{5})$: | $C_4$ | |
| This field is not Galois over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[\frac{3}{2}]$ | |
| Visible Swan slopes: | $[\frac{1}{2}]$ | |
| Means: | $\langle\frac{2}{5}\rangle$ | |
| Rams: | $(1)$ | |
| Jump set: | undefined | |
| Roots of unity: | $24 = (5^{ 2 } - 1)$ |
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Intermediate fields
| $\Q_{5}(\sqrt{2})$, 5.2.2.2a1.1, 5.1.5.6a1.1, 5.2.5.12a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{2} + 4 x + 2 \)
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| Relative Eisenstein polynomial: |
\( x^{10} + \left(10 t + 20\right) x^{4} + 5 t \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^5 + 2$,$2 z^4 + (3 t + 3)$ |
| Associated inertia: | $1$,$2$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $40$ |
| Galois group: | $C_4\times D_5$ (as 20T6) |
| 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 |