$( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{5} + \left(16 x + 2\right) ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 2$
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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: | $1$ |
| $\Aut(K/\Q_{2})$:
|
$C_2$ |
| 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: | $[1, 5, 13, 21]$ |
| Roots of unity: | $12 = (2^{ 2 } - 1) \cdot 2^{ 2 }$ |
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: |
\( x^{8} + \left(8 t + 8\right) x^{7} + 8 t x^{5} + 2 x^{4} + 16 x + 2 \)
$\ \in\Q_{2}(t)[x]$
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| Galois degree: |
$2048$
|
| Galois group: |
$(C_2^2\times D_4^2).D_4$ (as 16T1414)
|
| Inertia group: |
intransitive group not computed
|
| Wild inertia group: |
not computed
|
| Galois unramified degree: |
$4$
|
| Galois tame degree: |
$1$
|
| Galois Artin slopes: |
$[2, 2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]$
|
| Galois Swan slopes: |
$[1,1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]$
|
| Galois mean slope: |
$4.25390625$
|
| Galois splitting model: |
$x^{16} - 12 x^{14} + 28 x^{12} + 64 x^{10} + 1064 x^{8} - 7864 x^{6} + 13272 x^{4} - 208 x^{2} + 52$
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