Defining polynomial
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$( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$
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Invariants
| Base field: | $\Q_{2}$ |
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| Degree $d$: | $12$ |
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| Ramification index $e$: | $6$ |
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| Residue field degree $f$: | $2$ |
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| Discriminant exponent $c$: | $20$ |
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| Discriminant root field: | $\Q_{2}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{2})$: | $C_2^2$ | |
| This field is not Galois over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[\frac{8}{3}]$ | |
| Visible Swan slopes: | $[\frac{5}{3}]$ | |
| Means: | $\langle\frac{5}{6}\rangle$ | |
| Rams: | $(5)$ | |
| Jump set: | $[3, 9]$ | |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
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Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.1.3.2a1.1 x3, 2.2.3.4a1.2, 2.1.6.10a1.8, 2.1.6.10a1.7 |
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
\( x^{2} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{6} + 2 x^{5} + 4 x^{3} + 4 x + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^4 + z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 0]$ |
Invariants of the Galois closure
| Galois degree: | $24$ |
| Galois group: | $S_4$ (as 12T9) |
| Inertia group: | Intransitive group isomorphic to $A_4$ |
| Wild inertia group: | $C_2^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{8}{3}, \frac{8}{3}]$ |
| Galois Swan slopes: | $[\frac{5}{3},\frac{5}{3}]$ |
| Galois mean slope: | $2.1666666666666665$ |
| Galois splitting model: | $x^{12} + x^{10} + 2 x^{8} + 3 x^{6} + 2 x^{4} + x^{2} + 1$ |