Defining polynomial
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$( x^{3} + x + 1 )^{2} + 4 ( x^{3} + x + 1 ) + 10$
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Invariants
| Base field: | $\Q_{2}$ |
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| Degree $d$: | $6$ |
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| Ramification index $e$: | $2$ |
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| Residue field degree $f$: | $3$ |
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| Discriminant exponent $c$: | $9$ |
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| Discriminant root field: | $\Q_{2}(\sqrt{2\cdot 5})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{2})$ $=$ $\Gal(K/\Q_{2})$: | $C_6$ | |
| This field is Galois and abelian over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[3]$ | |
| Visible Swan slopes: | $[2]$ | |
| Means: | $\langle1\rangle$ | |
| Rams: | $(2)$ | |
| Jump set: | $[1, 3]$ | |
| Roots of unity: | $14 = (2^{ 3 } - 1) \cdot 2$ |
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Intermediate fields
| $\Q_{2}(\sqrt{2\cdot 5})$, 2.3.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.3.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{3} + x + 1 \)
|
|
| Relative Eisenstein polynomial: |
\( x^{2} + 4 x + 10 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + (t^2 + t + 1)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $6$ |
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_2$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[3]$ |
| Galois Swan slopes: | $[2]$ |
| Galois mean slope: | $1.5$ |
| Galois splitting model: | $x^{6} + 12 x^{4} + 36 x^{2} + 24$ |