Defining polynomial
\(x^{9} + 9 x^{8} + 45 x^{7} + 90 x^{6} + 324 x^{5} + 621 x^{4} + 594 x^{3} + 2349 x^{2} + 2943\) |
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: | \( x^{3} + \left(3 t + 3\right) x^{2} + 9 t + 12 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + t + 1$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
Galois group: | $C_3\wr C_3$ (as 9T17) |
Inertia group: | Intransitive group isomorphic to $C_3^3$ |
Wild inertia group: | $C_3^3$ |
Unramified degree: | $3$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 2, 2]$ |
Galois mean slope: | $52/27$ |
Galois splitting model: | $x^{9} - 597 x^{7} - 2587 x^{6} + 89550 x^{5} + 717594 x^{4} - 1413099 x^{3} - 28457199 x^{2} - 86897529 x - 72752609$ |