Defining polynomial
|
\(x^{12} + 4 x^{11} + 24 x^{10} - 8 x^{9} + 140 x^{8} - 1488 x^{7} + 832 x^{6} - 10208 x^{5} + 20336 x^{4} + 12160 x^{3} + 113280 x^{2} + 139520 x + 45376\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $6$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.3.0.1, 2.6.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.6.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{6} + x^{4} + x^{3} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 2 t^{4} x + 4 t^{5} + 4 t^{4} + 4 t^{2} + 6 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + t^{4}$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $D_4\times A_4$ (as 12T51) |
| Inertia group: | Intransitive group isomorphic to $C_2^4$ |
| Wild inertia group: | $C_2^4$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2, 2]$ |
| Galois mean slope: | $15/8$ |
| Galois splitting model: | $x^{12} - 4 x^{10} - 2 x^{8} + 9 x^{6} + 2 x^{4} - 4 x^{2} - 1$ |