Defining polynomial
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\(x^{12} + 2 x^{10} + 6 x^{9} + 6 x^{8} + 8 x^{7} + 28 x^{6} + 24 x^{5} + 20 x^{4} + 24 x^{3} + 32 x^{2} + 24 x + 8\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $3$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[4/3, 4/3]$ |
Intermediate fields
| 2.3.0.1, 2.4.4.5 |
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: |
\( x^{4} + \left(2 t^{2} + 2\right) x^{2} + 2 x + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times S_4$ (as 12T45) |
| Inertia group: | Intransitive group isomorphic to $A_4$ |
| Wild inertia group: | $C_2^2$ |
| Unramified degree: | $6$ |
| Tame degree: | $3$ |
| Wild slopes: | $[4/3, 4/3]$ |
| Galois mean slope: | $7/6$ |
| Galois splitting model: |
$x^{12} - 4 x^{9} + 54 x^{8} - 54 x^{7} + 122 x^{6} - 72 x^{5} + 93 x^{4} - 48 x^{3} + 18 x^{2} + 1$
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