Defining polynomial
\(x^{9} - 6 x^{8} - 36 x^{7} + 441 x^{6} + 126 x^{5} + 216 x^{4} + 189 x^{3} + 1404 x^{2} - 216\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $9$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $3$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
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: | \( x^{3} + \left(6 t^{2} + 6\right) x^{2} + 9 t + 3 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + 2t^{2} + 2$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
Galois group: | $C_3^2:C_6$ (as 9T13) |
Inertia group: | Intransitive group isomorphic to $C_3^2$ |
Wild inertia group: | $C_3^2$ |
Unramified degree: | $6$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 2]$ |
Galois mean slope: | $16/9$ |
Galois splitting model: | $x^{9} + 3 x^{7} - 6 x^{6} - 18 x^{5} - 12 x^{4} - 22 x^{3} - 27 x^{2} + 12 x + 13$ |