Defining polynomial
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$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{3} + 18 ( x^{2} + 2 x + 2 )^{2} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 ) + 3$
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{3})$: | $C_3$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[\frac{5}{2}]$ |
| Visible Swan slopes: | $[\frac{3}{2}]$ |
| Means: | $\langle1\rangle$ |
| Rams: | $(3)$ |
| Jump set: | $[1, 7]$ |
| Roots of unity: | $24 = (3^{ 2 } - 1) \cdot 3$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3\cdot 2})$, 3.2.2.2a1.2 |
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: |
\( x^{6} + 9 x^{3} + \left(18 t + 9\right) x^{2} + \left(9 t + 9\right) x + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^3 + 2$,$2 z^2 + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[6, 0]$ |
Invariants of the Galois closure
| Galois degree: | $324$ |
| Galois group: | $C_3\wr C_2^2$ (as 12T130) |
| Inertia group: | Intransitive group isomorphic to $C_3^2:C_6$ |
| Wild inertia group: | $C_3^3$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[\frac{3}{2}, 2, \frac{5}{2}]$ |
| Galois Swan slopes: | $[\frac{1}{2},1,\frac{3}{2}]$ |
| Galois mean slope: | $2.240740740740741$ |
| Galois splitting model: |
$x^{12} - 21 x^{9} + 399 x^{6} - 3528 x^{3} + 12348$
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