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: |
\( x^{3} + 11 \)
$\ \in\Q_{11}(t)[x]$
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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: | $x^{9} - 4 x^{8} + 32 x^{7} - 97 x^{6} + 331 x^{5} - 647 x^{4} + 965 x^{3} - 926 x^{2} + 345 x - 41$ |