Defining polynomial
|
$( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 18$
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $33$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-2})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[3, 4]$ |
| Visible Swan slopes: | $[2,3]$ |
| Means: | $\langle1, 2\rangle$ |
| Rams: | $(2, 4)$ |
| Jump set: | $[1, 3, 7]$ |
| Roots of unity: | $14 = (2^{ 3 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{2})$, 2.3.1.0a1.1, 2.3.2.9a1.5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.3.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{3} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + 4 x^{2} + 8 x + 18 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^2 + (t + 1)$,$(t + 1) z + (t^2 + 1)$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[8, 4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $96$ |
| Galois group: | $D_4\times A_4$ (as 12T51) |
| Inertia group: | Intransitive group isomorphic to $C_2^2\times D_4$ |
| Wild inertia group: | $C_2^2\times D_4$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 2, 3, 4]$ |
| Galois Swan slopes: | $[1,1,1,2,3]$ |
| Galois mean slope: | $3.1875$ |
| Galois splitting model: | $x^{12} - 10 x^{8} + 24 x^{4} - 8$ |