Defining polynomial
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$( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{5} + 6 x^{4} + 3 x + 3\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 ) + 3$
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Invariants
| Base field: | $\Q_{3}$ |
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| Degree $d$: | $18$ |
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| Ramification index $e$: | $3$ |
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| Residue field degree $f$: | $6$ |
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| Discriminant exponent $c$: | $18$ |
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| Discriminant root field: | $\Q_{3}(\sqrt{2})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{3})$: | $C_1$ | |
| 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: | $(\frac{1}{2})$ | |
| Jump set: | undefined | |
| Roots of unity: | $728 = (3^{ 6 } - 1)$ |
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Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.3.1.0a1.1, 3.6.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 3.6.1.0a1.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 \)
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|
| Relative Eisenstein polynomial: |
\( x^{3} + \left(3 t^{5} + 6 t^{4} + 3 t + 3\right) x + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + (2 t^5 + t^4 + 2 t^2)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $972$ |
| Galois group: | $C_3^3:S_3^2$ (as 18T234) |
| 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 |