Defining polynomial
|
\(x^{6} + 6 x^{3} + 81 x^{2} + 9\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $6$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[5/2]$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{2} + 2 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + \left(9 t + 9\right) x + 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
| Galois group: | $S_3^2$ (as 6T9) |
| Inertia group: | Intransitive group isomorphic to $C_3:S_3$ |
| Wild inertia group: | $C_3^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $2$ |
| Wild slopes: | $[3/2, 5/2]$ |
| Galois mean slope: | $37/18$ |
| Galois splitting model: |
$x^{6} + 3 x^{3} + 72$
|