Defining polynomial
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\(x^{12} + 2 x^{10} - 2 x^{9} + 8 x^{8} + 4 x^{7} + 12 x^{6} - 4 x^{5} + 8 x^{4} - 4 x^{3} + 8 x^{2} + 4\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $16$ |
| 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
| $\Q_{2}(\sqrt{5})$, 2.3.2.1 x3, 2.6.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{6} + \left(2 t + 2\right) x^{4} + 2 t x^{3} + 2 x^{2} + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + t$,$z^{4} + z^{2} + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
| Galois group: | $\GL(2,\mathbb{Z}/4)$ (as 12T50) |
| Inertia group: | Intransitive group isomorphic to $C_2^2\times A_4$ |
| Wild inertia group: | $C_2^4$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | $[4/3, 4/3, 2, 2]$ |
| Galois mean slope: | $43/24$ |
| Galois splitting model: |
$x^{12} - 4 x^{11} + 13 x^{10} - 28 x^{9} + 50 x^{8} - 70 x^{7} + 71 x^{6} - 42 x^{5} + 6 x^{4} + 10 x^{3} - 7 x^{2} + 1$
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