Defining polynomial
|
\(x^{12} + 12 x^{11} + 58 x^{10} + 116 x^{9} + 374 x^{8} + 648 x^{7} + 1472 x^{6} + 1584 x^{5} + 3036 x^{4} + 2816 x^{3} + 3112 x^{2} + 1776 x + 2088\)
|
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, 2.6.6.1 |
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
\( x^{3} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + \left(4 t + 4\right) x^{3} + \left(6 t^{2} + 2 t + 2\right) x^{2} + \left(4 t + 4\right) x + 8 t^{2} + 8 t + 10 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $(t^{2} + t + 1)z + t + 1$,$z^{2} + t^{2} + t + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^4.A_4$ (as 12T92) |
| Inertia group: | Intransitive group isomorphic to $C_2\times C_4^2$ |
| Wild inertia group: | $C_2\times C_4^2$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 3, 3, 3]$ |
| Galois mean slope: | $45/16$ |
| Galois splitting model: | $x^{12} - 9 x^{8} + 20 x^{6} - 24 x^{4} + 18 x^{2} - 3$ |