Defining polynomial
|
\(x^{9} + 6 x^{7} + 60 x^{6} + 12 x^{5} + 42 x^{4} - 1465 x^{3} + 240 x^{2} - 1560 x + 8088\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 11 }) }$: | $3$ |
| This field is not Galois over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| 11.3.2.1, 11.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 11.3.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of
\( x^{3} + 2 x + 9 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + 11 \)
$\ \in\Q_{11}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{2} + 3z + 3$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times S_3$ (as 9T4) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | $x^{9} - 4 x^{8} + 32 x^{7} - 97 x^{6} + 331 x^{5} - 647 x^{4} + 965 x^{3} - 926 x^{2} + 345 x - 41$ |