Defining polynomial
|
\(x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $6$ |
| Discriminant exponent $c$: | $18$ |
| 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: | $[3]$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.3.0.1, 2.4.6.2, 2.6.0.1, 2.6.9.5, 2.6.9.7 |
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: |
\( x^{2} + \left(4 t^{5} + 4 t^{4} + 4 t^{3} + 4 t^{2}\right) x + 8 t^{5} + 4 t^{3} + 4 t^{2} + 8 t + 14 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_6$ (as 12T2) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_2$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | $[3]$ |
| Galois mean slope: | $3/2$ |
| Galois splitting model: | $x^{12} - 8 x^{6} + 64$ |