Defining polynomial
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$( x^{2} + 2 x + 2 )^{9} + \left(15 x + 24\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{5} + 9 ( x^{2} + 2 x + 2 )^{4} + 3$
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $18$ |
| Ramification index $e$: | $9$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $42$ |
| 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: | $[2, \frac{17}{6}]$ |
| Visible Swan slopes: | $[1,\frac{11}{6}]$ |
| Means: | $\langle\frac{2}{3}, \frac{13}{9}\rangle$ |
| Rams: | $(1, \frac{7}{2})$ |
| Jump set: | undefined |
| Roots of unity: | $8 = (3^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.2.3.8a3.1 |
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: |
\( x^{9} + 18 x^{7} + 3 t x^{6} + 18 x^{5} + 9 x^{4} + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^6 + 2 t$,$2 t z + (2 t + 2)$ |
| Associated inertia: | $2$,$1$ |
| Indices of inseparability: | $[13, 6, 0]$ |
Invariants of the Galois closure
| Galois degree: | $52488$ |
| Galois group: | $(C_3\wr C_3)^2:(C_2\times C_4)$ (as 18T727) |
| 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 |