Defining polynomial
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$( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 ) + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $16$ |
| 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: | $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}]$ |
| Visible Swan slopes: | $[\frac{1}{7},\frac{1}{7},\frac{1}{7}]$ |
| Means: | $\langle\frac{1}{14}, \frac{3}{28}, \frac{1}{8}\rangle$ |
| Rams: | $(\frac{1}{7}, \frac{1}{7}, \frac{1}{7})$ |
| Jump set: | $[1, 2, 4, 9]$ |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.1.8.8a1.1 |
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: |
\( x^{8} + 2 x + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 1, 1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $336$ |
| Galois group: | $F_8:C_6$ (as 16T712) |
| Inertia group: | Intransitive group isomorphic to $F_8$ |
| Wild inertia group: | $C_2^3$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $7$ |
| Galois Artin slopes: | $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}]$ |
| Galois Swan slopes: | $[\frac{1}{7},\frac{1}{7},\frac{1}{7}]$ |
| Galois mean slope: | $1.1071428571428572$ |
| Galois splitting model: |
$x^{16} - 4 x^{15} + 2 x^{14} - 28 x^{13} + 126 x^{12} - 56 x^{11} + 140 x^{10} - 1038 x^{9} + 407 x^{8} + 752 x^{7} + 588 x^{6} + 434 x^{5} + 238 x^{4} - 84 x^{3} - 10 x^{2} - 2 x + 1$
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