Defining polynomial
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\(x^{12} + 12 x^{11} + 46 x^{10} + 52 x^{9} + 58 x^{8} + 320 x^{7} + 288 x^{6} + 944 x^{5} + 724 x^{4} + 1168 x^{3} + 536 x^{2} + 432 x + 72\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $24$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2, 3]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, 2.3.0.1, 2.6.6.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{3} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{4} + 4 x^{3} + \left(4 t^{2} + 4 t + 2\right) x^{2} + \left(4 t^{2} + 4\right) x + 4 t^{2} + 4 t + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + t^{2} + 1$,$z^{2} + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\wr C_6$ (as 12T134) |
| Inertia group: | Intransitive group isomorphic to $C_2^3:D_4$ |
| Wild inertia group: | $C_2^3:D_4$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2, 2, 3, 3]$ |
| Galois mean slope: | $87/32$ |
| Galois splitting model: |
$x^{12} + 10 x^{10} + 30 x^{8} + 20 x^{6} - 45 x^{4} - 72 x^{2} - 27$
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