Defining polynomial
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$( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 4 x + 10$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $40$ |
| 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: | $[2, 3, 3]$ |
| Visible Swan slopes: | $[1,2,2]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{13}{8}\rangle$ |
| Rams: | $(1, 3, 3)$ |
| Jump set: | $[1, 2, 4, 16]$ |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.2.2.4a1.2, 2.2.4.16b2.10, 2.2.4.16b1.6, 2.2.4.16b2.8 |
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: |
\( x^{8} + 4 x^{5} + 2 x^{4} + 4 t + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^4 + 1$,$z^3 + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[13, 12, 4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $128$ |
| Galois group: | $C_2^3:\OD_{16}$ (as 16T252) |
| Inertia group: | Intransitive group isomorphic to $C_2^3\times C_4$ |
| Wild inertia group: | $C_2^3\times C_4$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 2, 3, 3]$ |
| Galois Swan slopes: | $[1,1,1,2,2]$ |
| Galois mean slope: | $2.6875$ |
| Galois splitting model: |
$x^{16} - 4 x^{14} - 18 x^{12} - 132 x^{10} - 525 x^{8} - 1108 x^{6} - 1358 x^{4} + 864 x^{2} + 361$
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