$( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + \left(2 x + 2\right) ( x^{2} + x + 1 )^{6} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 4 x + 10$
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| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $36$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$:
|
$C_2^2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[2, 2, 3]$ |
| Visible Swan slopes: | $[1,1,2]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{4}, \frac{11}{8}\rangle$ |
| Rams: | $(1, 1, 5)$ |
| Jump set: | $[1, 3, 6, 16]$ |
| 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: |
\( x^{8} + 4 x^{7} + 2 t x^{6} + 4 x^{5} + 2 t x^{4} + 4 x^{3} + 4 t + 2 \)
$\ \in\Q_{2}(t)[x]$
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