Defining polynomial
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\(x^{12} + 10 x^{11} - 16 x^{10} - 316 x^{9} - 364 x^{8} + 1040 x^{7} + 7648 x^{6} + 17632 x^{5} + 25008 x^{4} + 17056 x^{3} + 6016 x^{2} + 1344 x + 4544\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $6$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.3.0.1, 2.6.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.6.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{6} + x^{4} + x^{3} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{2} + \left(2 t^{5} + 2 t^{4} + 2 t^{3} + 2\right) x + 4 t^{5} + 4 t^{3} + 4 t^{2} + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + t^{5} + t^{4} + t^{3} + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\wr C_6$ (as 12T134) |
| Inertia group: | Intransitive group isomorphic to $C_2^6$ |
| Wild inertia group: | $C_2^6$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2, 2, 2, 2]$ |
| Galois mean slope: | $63/32$ |
| Galois splitting model: |
$x^{12} - 52 x^{8} + 156 x^{6} - 169 x^{4} + 78 x^{2} - 13$
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