Defining polynomial
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$( x^{3} + 2 x + 1 )^{6} + 6 ( x^{3} + 2 x + 1 )^{5} + 6 x ( x^{3} + 2 x + 1 )^{4} + 3 ( x^{3} + 2 x + 1 ) + 3 x$
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $18$ |
| Ramification index $e$: | $6$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $27$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{3})$: | $C_3$ |
| Visible Artin slopes: | $[2]$ |
| Visible Swan slopes: | $[1]$ |
| Means: | $\langle\frac{2}{3}\rangle$ |
| Rams: | $(2)$ |
| Jump set: | undefined |
| Roots of unity: | $26 = (3^{ 3 } - 1)$ |
Intermediate fields
| $\Q_{3}(\sqrt{3})$, 3.3.1.0a1.1, 3.3.2.3a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 3.3.1.0a1.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{3} + 2 x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{6} + \left(3 t^{2} + 3 t + 3\right) x^{5} + \left(6 t^{2} + 6 t + 3\right) x^{4} + 3 t \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^3 + 2$,$2 z^2 + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois degree: | not computed |
| Galois group: | not computed |
| 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 |