Defining polynomial
|
\(x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $6$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-2})$ |
| Root number: | $-i$ |
| $\card{ \Gal(K/\Q_{ 2 }) }$: | $6$ |
| This field is Galois and abelian over $\Q_{2}.$ | |
| Visible slopes: | $[3]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-2})$, 2.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{3} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 4 x + 6 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_2$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | $[3]$ |
| Galois mean slope: | $3/2$ |
| Galois splitting model: | $x^{6} + 12 x^{4} + 36 x^{2} + 8$ |