Defining polynomial
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\(x^{12} + 6 x^{10} + 4 x^{9} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $28$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $4$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[8/3, 3]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-2})$, 2.3.2.1, 2.6.10.6, 2.6.11.4, 2.6.11.9 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{12} + 6 x^{10} + 4 x^{9} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 2 \)
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Ramification polygon
| Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{8} + z^{4} + 1$ |
| Associated inertia: | $1$,$1$,$2$ |
| Indices of inseparability: | $[17, 10, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^2\times S_4$ (as 12T48) |
| Inertia group: | $C_2^2\times A_4$ (as 12T25) |
| Wild inertia group: | $C_2^4$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | $[2, 8/3, 8/3, 3]$ |
| Galois mean slope: | $8/3$ |
| Galois splitting model: |
$x^{12} + 4 x^{10} - 12 x^{9} + 15 x^{8} - 24 x^{7} + 56 x^{6} - 92 x^{5} + 101 x^{4} - 72 x^{3} + 32 x^{2} - 8 x + 1$
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