Defining polynomial
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$( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $36$ |
| 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: | $[1, 2, 7, 15]$ |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.2.2.6a1.1, 2.1.4.6a2.1, 2.2.4.12a3.1, 2.1.8.18b2.2, 2.1.8.18b2.1 |
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
\( x^{2} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{8} + 2 x^{6} + 2 x^{4} + 4 x^{3} + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^6 + z^2 + 1$,$z + 1$ |
| Associated inertia: | $3$,$1$ |
| Indices of inseparability: | $[11, 6, 4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $48$ |
| Galois group: | $C_2^2\times A_4$ (as 16T58) |
| Inertia group: | Intransitive group isomorphic to $C_2^3$ |
| Wild inertia group: | $C_2^3$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 3]$ |
| Galois Swan slopes: | $[1,1,2]$ |
| Galois mean slope: | $2.25$ |
| Galois splitting model: |
$x^{16} - 12 x^{14} + 38 x^{12} + 72 x^{10} + 127 x^{8} + 152 x^{6} + 106 x^{4} + 44 x^{2} + 9$
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