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: | $x^{9} - 24 x^{7} - 23 x^{6} + 135 x^{5} + 159 x^{4} - 209 x^{3} - 180 x^{2} + 141 x + 11$ |