Defining polynomial
|
\(x^{12} - 8 x^{11} + 78 x^{10} + 116 x^{9} - 14 x^{8} - 48 x^{7} + 1504 x^{6} + 3216 x^{5} + 2716 x^{4} + 2880 x^{3} + 3160 x^{2} + 1904 x + 1016\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $24$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $4$ |
| 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: |
\( x^{4} + \left(4 t^{2} + 4 t\right) x^{3} + \left(4 t^{2} + 6 t + 2\right) x^{2} + \left(4 t + 4\right) x + 8 t^{2} + 14 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $(t + 1)z + t + 1$,$z^{2} + t + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_4^2:C_6$ (as 12T55) |
| Inertia group: | Intransitive group isomorphic to $C_2\times C_4^2$ |
| Wild inertia group: | $C_2\times C_4^2$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2, 3, 3]$ |
| Galois mean slope: | $43/16$ |
| Galois splitting model: |
$x^{12} - 6 x^{10} - 23 x^{8} + 210 x^{6} - 360 x^{4} + 50 x^{2} + 25$
|