Defining polynomial
|
\(x^{9} + 6 x^{8} + 45 x^{7} + 252 x^{6} + 144 x^{5} + 540 x^{4} + 432 x^{3} + 864 x^{2} + 1728\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $3$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| 3.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.3.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{3} + 2 x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + \left(3 t^{2} + 3 t + 6\right) x^{2} + 12 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{2} + t^{2} + t + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois group: | $\He_3$ (as 9T7) |
| Inertia group: | Intransitive group isomorphic to $C_3^2$ |
| Wild inertia group: | $C_3^2$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2]$ |
| Galois mean slope: | $16/9$ |
| Galois splitting model: | $x^{9} - 24 x^{7} - 23 x^{6} + 135 x^{5} + 159 x^{4} - 209 x^{3} - 180 x^{2} + 141 x + 11$ |