$( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 24 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$
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| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $56$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$:
|
$C_2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[3, 4, \frac{17}{4}]$ |
| Visible Swan slopes: | $[2,3,\frac{13}{4}]$ |
| Means: | $\langle1, 2, \frac{21}{8}\rangle$ |
| Rams: | $(2, 4, 5)$ |
| 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: |
\( x^{8} + 8 t x^{7} + 8 x^{5} + 4 t x^{4} + \left(8 t + 8\right) x^{2} + 16 t x + 8 t + 2 \)
$\ \in\Q_{2}(t)[x]$
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| Galois degree: |
$512$
|
| Galois group: |
$C_2^5.\SD_{16}$ (as 16T955)
|
| Inertia group: |
Intransitive group isomorphic to $C_2^3.C_4^2$
|
| Wild inertia group: |
$C_2^3.C_4^2$
|
| Galois unramified degree: |
$4$
|
| Galois tame degree: |
$1$
|
| Galois Artin slopes: |
$[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}]$
|
| Galois Swan slopes: |
$[1,2,2,\frac{5}{2},3,3,\frac{13}{4}]$
|
| Galois mean slope: |
$4.0$
|
| Galois splitting model: |
$x^{16} - 28 x^{14} + 238 x^{12} - 240 x^{11} - 856 x^{10} + 2280 x^{9} + 95 x^{8} - 6720 x^{7} + 4864 x^{6} + 7400 x^{5} - 9242 x^{4} - 1280 x^{3} + 6812 x^{2} - 4040 x + 671$
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