Defining polynomial
|
$( x^{2} + x + 1 )^{8} + \left(2 x + 2\right) ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $24$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_1$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}]$ |
| Visible Swan slopes: | $[\frac{5}{7},\frac{5}{7},\frac{5}{7}]$ |
| Means: | $\langle\frac{5}{14}, \frac{15}{28}, \frac{5}{8}\rangle$ |
| Rams: | $(\frac{5}{7}, \frac{5}{7}, \frac{5}{7})$ |
| Jump set: | $[1, 3, 6, 13]$ |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{8} + 2 t x^{5} + 2 t x^{4} + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + t$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[5, 5, 4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $2688$ |
| Galois group: | $C_2^3:F_8:C_6$ (as 16T1501) |
| Inertia group: | not computed |
| Wild inertia group: | not computed |
| Galois unramified degree: | not computed |
| Galois tame degree: | not computed |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: | not computed |