Defining polynomial
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\(x^{12} + 12 x^{10} + 32 x^{9} + 84 x^{8} + 32 x^{7} + 544 x^{6} - 704 x^{5} - 4176 x^{4} - 5248 x^{3} + 16064 x^{2} + 21248 x + 6592\)
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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: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $4$ |
| 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: |
\( x^{2} + \left(2 t^{4} + 2 t^{2} + 2 t\right) x + 4 t^{5} + 4 t^{3} + 4 t + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + t^{4} + t^{2} + t$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_4\times A_4$ (as 12T29) |
| Inertia group: | Intransitive group isomorphic to $C_2^2$ |
| Wild inertia group: | $C_2^2$ |
| Unramified degree: | $12$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2]$ |
| Galois mean slope: | $3/2$ |
| Galois splitting model: |
$x^{12} - 13 x^{10} + 52 x^{8} - 52 x^{6} - 91 x^{4} + 143 x^{2} + 13$
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