Defining polynomial
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\(x^{12} + 4 x^{10} + 24 x^{9} + 2 x^{8} - 104 x^{7} - 224 x^{6} - 976 x^{5} - 492 x^{4} - 2448 x^{3} - 768 x^{2} - 1600 x - 1048\)
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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
| 2.3.0.1, 2.6.6.6 |
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: |
\( x^{4} + 4 t x^{3} + \left(6 t^{2} + 2 t\right) x^{2} + 4 t x + 8 t^{2} + 12 t + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $(t^{2} + t)z + t$,$z^{2} + t^{2} + t$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_4^3:C_6$ (as 12T141) |
| Inertia group: | Intransitive group isomorphic to $C_4^2:C_2^2$ |
| Wild inertia group: | $C_4^2:C_2^2$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2, 2, 3, 3]$ |
| Galois mean slope: | $87/32$ |
| Galois splitting model: |
$x^{12} + 12 x^{10} - 42 x^{8} + 60 x^{6} - 45 x^{4} + 18 x^{2} - 3$
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