Defining polynomial
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$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $20$ |
| Ramification index $e$: | $10$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $38$ |
| 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: | $[3]$ |
| Visible Swan slopes: | $[2]$ |
| Means: | $\langle1\rangle$ |
| Rams: | $(10)$ |
| Jump set: | $[5, 15]$ |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.1.5.4a1.1, 2.2.5.8a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{10} + 4 t x^{9} + 4 t x^{5} + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^8 + z^6 + 1$,$z + 1$ |
| Associated inertia: | $2$,$1$ |
| Indices of inseparability: | $[10, 0]$ |
Invariants of the Galois closure
| Galois degree: | $1280$ |
| Galois group: | $C_2^6:F_5$ (as 20T192) |
| 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 |