Defining polynomial
|
$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x + 12\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + 8 ( x^{4} + x + 1 ) + 2$
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $44$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$: | $C_2^2$ |
| 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: | $30 = (2^{ 4 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.4.1.0a1.1, 2.2.2.6a1.3, 2.2.2.6a1.4, 2.4.2.12a1.7, 2.2.4.22a1.33, 2.2.4.22a1.36 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.4.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{4} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + \left(8 t^{3} + 8\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + 8 x + 16 t^{3} + 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[8, 4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $128$ |
| Galois group: | $C_2^5:C_4$ (as 16T227) |
| Inertia group: | Intransitive group isomorphic to $C_2^2\wr C_2$ |
| Wild inertia group: | $C_2^2\wr C_2$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 3, \frac{7}{2}, 4]$ |
| Galois Swan slopes: | $[1,1,2,\frac{5}{2},3]$ |
| Galois mean slope: | $3.4375$ |
| Galois splitting model: |
$x^{16} - 4 x^{14} + 16 x^{12} - 64 x^{10} + 136 x^{8} - 320 x^{6} + 440 x^{4} + 400$
|