$( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{6} + \left(4 x + 14\right) ( x^{2} + x + 1 )^{5} + \left(2 x + 8\right) ( x^{2} + x + 1 )^{4} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 2$
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| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $54$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$:
|
$C_4$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[2, \frac{7}{2}, \frac{9}{2}]$ |
| Visible Swan slopes: | $[1,\frac{5}{2},\frac{7}{2}]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{2}, \frac{5}{2}\rangle$ |
| Rams: | $(1, 4, 8)$ |
| Jump set: | $[1, 3, 7, 15]$ |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{8} + 8 t x^{7} + \left(2 t + 8\right) x^{4} + 8 x^{2} + 16 x + 6 \)
$\ \in\Q_{2}(t)[x]$
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| Galois degree: |
$1024$
|
| Galois group: |
$D_4^2:\OD_{16}$ (as 16T1120)
|
| Inertia group: |
Intransitive group isomorphic to $D_4^2:C_4$
|
| Wild inertia group: |
$D_4^2:C_4$
|
| Galois unramified degree: |
$4$
|
| Galois tame degree: |
$1$
|
| Galois Artin slopes: |
$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{9}{2}, \frac{9}{2}]$
|
| Galois Swan slopes: |
$[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]$
|
| Galois mean slope: |
$4.2734375$
|
| Galois splitting model: |
$x^{16} - 80 x^{12} - 40 x^{10} - 810 x^{8} + 720 x^{6} - 300 x^{4} - 240 x^{2} + 20$
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