Defining polynomial
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\(x^{12} + 42 x^{11} + 747 x^{10} + 7280 x^{9} + 41955 x^{8} + 143682 x^{7} + 279531 x^{6} + 287826 x^{5} + 175245 x^{4} + 124460 x^{3} + 344757 x^{2} + 893466 x + 996620\)
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Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{11}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 11 }) }$: | $12$ |
| This field is Galois over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{11}(\sqrt{2})$, $\Q_{11}(\sqrt{11})$, $\Q_{11}(\sqrt{11\cdot 2})$, 11.3.2.1 x3, 11.4.2.1, 11.6.4.1, 11.6.5.1 x3, 11.6.5.2 x3 |
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: |
\( x^{6} + 11 \)
$\ \in\Q_{11}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^{5} + 6z^{4} + 4z^{3} + 9z^{2} + 4z + 6$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $D_6$ (as 12T3) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | None |
| Galois mean slope: | $5/6$ |
| Galois splitting model: |
$x^{12} + 3146 x^{6} + 14235529$
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