Defining polynomial
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$( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{2} + 2$
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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: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, 2]$ |
| Visible Swan slopes: | $[\frac{1}{3},\frac{1}{3},1]$ |
| Means: | $\langle\frac{1}{6}, \frac{1}{4}, \frac{5}{8}\rangle$ |
| Rams: | $(\frac{1}{3}, \frac{1}{3}, 3)$ |
| Jump set: | $[1, 2, 5, 13]$ |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.2.2.4a2.2, 2.1.4.4a1.1, 2.2.4.8a1.1 |
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: |
\( x^{8} + 2 t x^{7} + 2 t x^{5} + 2 x^{2} + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + 1$,$z + t$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 2, 2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $96$ |
| Galois group: | $\GL(2,\mathbb{Z}/4)$ (as 16T193) |
| Inertia group: | Intransitive group isomorphic to $C_2^2\times A_4$ |
| Wild inertia group: | $C_2^4$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, 2, 2]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3},1,1]$ |
| Galois mean slope: | $1.7916666666666667$ |
| Galois splitting model: |
$x^{16} - 110 x^{14} - 116 x^{13} + 3298 x^{12} + 4168 x^{11} - 37682 x^{10} - 56900 x^{9} + 153976 x^{8} + 298400 x^{7} - 40010 x^{6} - 316060 x^{5} - 147234 x^{4} + 41144 x^{3} + 46842 x^{2} + 11988 x + 999$
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