Defining polynomial
|
\(x^{12} - 8 x^{11} + 14 x^{10} + 76 x^{9} + 138 x^{8} + 432 x^{7} + 688 x^{6} + 992 x^{5} + 1748 x^{4} + 1728 x^{3} + 1848 x^{2} + 1648 x + 968\)
|
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: | $-1$ |
| $\card{ \Gal(K/\Q_{ 2 }) }$: | $12$ |
| This field is Galois and abelian over $\Q_{2}.$ | |
| Visible slopes: | $[2, 3]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{2})$, $\Q_{2}(\sqrt{-2})$, 2.3.0.1, 2.4.8.2, 2.6.6.3, 2.6.9.1, 2.6.9.5 |
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
\( x^{3} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + 4 t^{2} x^{3} + \left(4 t^{2} + 2\right) x^{2} + 4 x + 12 t^{2} + 8 t + 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{2} + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_6$ (as 12T2) |
| Inertia group: | Intransitive group isomorphic to $C_2^2$ |
| Wild inertia group: | $C_2^2$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 3]$ |
| Galois mean slope: | $2$ |
| Galois splitting model: | $x^{12} + 13 x^{8} + 26 x^{4} + 1$ |