Defining polynomial
|
\(x^{6} + 18 x^{4} + 6 x^{3} + 162 x^{2} + 216 x + 90\)
|
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: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $2$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[5/2]$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.3.5.3 |
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: |
\( x^{3} + \left(9 t + 18\right) x + 9 t + 12 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
| Galois group: | $D_6$ (as 6T3) |
| Inertia group: | Intransitive group isomorphic to $S_3$ |
| Wild inertia group: | $C_3$ |
| Unramified degree: | $2$ |
| Tame degree: | $2$ |
| Wild slopes: | $[5/2]$ |
| Galois mean slope: | $11/6$ |
| Galois splitting model: | $x^{6} - 18$ |