Defining polynomial
\(x^{6} - 110 x^{3} - 16819\)
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Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $4$ |
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
$\Q_{11}(\sqrt{2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of
\( x^{2} + 7 x + 2 \)
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Relative Eisenstein polynomial: |
\( x^{3} + 44 t + 99 \)
$\ \in\Q_{11}(t)[x]$
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Ramification polygon
Residual polynomials: | $z^{2} + 3z + 3$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_3\times S_3$ (as 6T5) |
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: |
$x^{6} - 11 x^{3} + 847$
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