Defining polynomial
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$( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{8} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{4} + 3 x ( x^{2} + 2 x + 2 )^{3} + 21$
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $18$ |
| Ramification index $e$: | $9$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $30$ |
| 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: | $[\frac{3}{2}, 2]$ |
| Visible Swan slopes: | $[\frac{1}{2},1]$ |
| Means: | $\langle\frac{1}{3}, \frac{7}{9}\rangle$ |
| Rams: | $(\frac{1}{2}, 2)$ |
| Jump set: | undefined |
| Roots of unity: | $8 = (3^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.2.3.6a5.1 |
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: |
\( x^{9} + \left(3 t + 6\right) x^{8} + 6 t x^{7} + \left(6 t + 3\right) x^{3} + 21 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^3 + (t + 2)$,$(t + 2) z^2 + 2 t$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[7, 3, 0]$ |
Invariants of the Galois closure
| Galois degree: | $108$ |
| Galois group: | $\He_3:C_4$ (as 18T49) |
| 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 |