Defining polynomial
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$( x^{2} + 2 x + 2 )^{6} + 6 ( x^{2} + 2 x + 2 )^{3} + 3 x ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$
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Invariants
| Base field: | $\Q_{3}$ |
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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 $c$: | $14$ |
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| Discriminant root field: | $\Q_{3}(\sqrt{2})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{3})$: | $S_3$ | |
| This field is not Galois over $\Q_{3}.$ | ||
| Visible Artin slopes: | $[\frac{3}{2}]$ | |
| Visible Swan slopes: | $[\frac{1}{2}]$ | |
| Means: | $\langle\frac{1}{3}\rangle$ | |
| Rams: | $(1)$ | |
| Jump set: | undefined | |
| Roots of unity: | $8 = (3^{ 2 } - 1)$ |
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Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.2.2.2a1.1, 3.2.3.6a5.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{2} + 2 x + 2 \)
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| Relative Eisenstein polynomial: |
\( x^{6} + 3 t x^{2} + 3 t \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^3 + 2$,$2 z^2 + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $36$ |
| Galois group: | $C_3^2:C_4$ (as 12T17) |
| Inertia group: | Intransitive group isomorphic to $C_3:S_3$ |
| Wild inertia group: | $C_3^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[\frac{3}{2}, \frac{3}{2}]$ |
| Galois Swan slopes: | $[\frac{1}{2},\frac{1}{2}]$ |
| Galois mean slope: | $1.3888888888888888$ |
| Galois splitting model: | $x^{12} + 12 x^{10} - 8 x^{9} + 78 x^{8} - 72 x^{7} + 308 x^{6} - 288 x^{5} + 711 x^{4} - 592 x^{3} + 924 x^{2} - 816 x + 526$ |