Defining polynomial
|
$( x^{4} + x + 1 )^{4} + \left(4 x + 6\right) ( x^{4} + x + 1 )^{3} + \left(2 x^{2} + 10 x\right) ( x^{4} + x + 1 )^{2} + 2$
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $36$ |
| 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: | $[2, \frac{7}{2}]$ |
| Visible Swan slopes: | $[1,\frac{5}{2}]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{2}\rangle$ |
| Rams: | $(1, 4)$ |
| 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, 2.2.2.4a2.2, 2.2.2.4a2.1, 2.4.2.8a3.1, 2.2.4.18a2.5, 2.2.4.18a2.6 |
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: |
\( x^{4} + 4 t^{2} x^{3} + \left(2 t^{2} + 2 t\right) x^{2} + \left(8 t^{2} + 8\right) x + 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^2 + (t^2 + t)$,$(t^2 + t) z + (t^2 + t)$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[6, 2, 0]$ |
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: | $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]$ |
| Galois Swan slopes: | $[1,1,2,\frac{5}{2},\frac{5}{2}]$ |
| Galois mean slope: | $3.1875$ |
| Galois splitting model: |
$x^{16} - 10 x^{14} + 60 x^{12} - 240 x^{10} + 640 x^{8} - 1200 x^{6} + 1400 x^{4} - 800 x^{2} + 400$
|