$( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + \left(12 x + 6\right) ( x^{2} + x + 1 )^{5} + \left(2 x + 24\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{2} + 16 x ( x^{2} + x + 1 ) + 2$
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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: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$:
|
$C_2$ |
| 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: |
\( x^{8} + 8 t x^{7} + 4 t x^{6} + \left(8 t + 8\right) x^{5} + \left(2 t + 24\right) x^{4} + 8 t x^{2} + 16 t x + 6 \)
$\ \in\Q_{2}(t)[x]$
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| Galois degree: |
$2048$
|
| Galois group: |
$C_2^5.C_2\wr C_2^2$ (as 16T1455)
|
| Inertia group: |
intransitive group not computed
|
| Wild inertia group: |
not computed
|
| Galois unramified degree: |
$2$
|
| Galois tame degree: |
$1$
|
| Galois Artin slopes: |
$[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, 4, 4, \frac{9}{2}, \frac{9}{2}]$
|
| Galois Swan slopes: |
$[1,1,2,2,\frac{5}{2},\frac{5}{2},3,3,\frac{7}{2},\frac{7}{2}]$
|
| Galois mean slope: |
$4.330078125$
|
| Galois splitting model: |
$x^{16} + 16 x^{14} + 112 x^{12} - 320 x^{10} - 3170 x^{8} + 8008 x^{6} + 26620 x^{4} - 85184 x^{2} + 58564$
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