$( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{2} + 24 ( x^{2} + x + 1 ) + 8 x + 2$
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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: |
\( x^{8} + 8 t x^{7} + 8 x^{6} + 8 t x^{5} + 4 t x^{4} + 8 x^{2} + 16 x + 8 t + 2 \)
$\ \in\Q_{2}(t)[x]$
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| Galois degree: |
$2048$
|
| Galois group: |
$C_2^7.(C_2\times D_4)$ (as 16T1413)
|
| Inertia group: |
intransitive group not computed
|
| Wild inertia group: |
not computed
|
| Galois unramified degree: |
$4$
|
| Galois tame degree: |
$1$
|
| Galois Artin slopes: |
$[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]$
|
| Galois Swan slopes: |
$[1,1,2,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]$
|
| Galois mean slope: |
$4.09765625$
|
| Galois splitting model: |
$x^{16} + 20 x^{14} + 130 x^{12} + 120 x^{10} - 865 x^{8} + 1200 x^{6} - 650 x^{4} + 100 x^{2} + 25$
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