Defining polynomial
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\(x^{12} + 2 x^{11} + 22 x^{10} + 52 x^{9} + 74 x^{8} + 56 x^{7} + 176 x^{6} + 168 x^{5} + 204 x^{4} + 72 x^{3} + 216 x^{2} + 216\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $18$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $1$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2, 2]$ |
Intermediate fields
| 2.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{3} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{4} + \left(2 t^{2} + 2 t + 2\right) x^{3} + \left(2 t + 2\right) x^{2} + 6 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^{3} + (t + 1)z + t^{2} + t + 1$ |
| Associated inertia: | $3$ |
| Indices of inseparability: | $[3, 2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^6:C_9$ (as 12T166) |
| Inertia group: | Intransitive group isomorphic to $C_2^6$ |
| Wild inertia group: | $C_2^6$ |
| Unramified degree: | $9$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2, 2, 2, 2]$ |
| Galois mean slope: | $63/32$ |
| Galois splitting model: |
$x^{12} - 4 x^{11} - 20 x^{10} + 112 x^{9} + x^{8} - 782 x^{7} + 1254 x^{6} + 546 x^{5} - 3484 x^{4} + 4144 x^{3} - 2364 x^{2} + 674 x - 77$
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