Defining polynomial
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$( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 ) + 8 x + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $52$ |
| 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: | $[3, \frac{7}{2}, 4]$ |
| Visible Swan slopes: | $[2,\frac{5}{2},3]$ |
| Means: | $\langle1, \frac{7}{4}, \frac{19}{8}\rangle$ |
| Rams: | $(2, 3, 5)$ |
| Jump set: | $[1, 3, 7, 15]$ |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.2.2.6a1.2, 2.2.4.20a1.18, 2.2.4.22a1.17, 2.2.4.22a1.24 |
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: |
\( x^{8} + 8 t x^{7} + 4 x^{6} + \left(8 t + 4\right) x^{4} + 8 x^{3} + 8 t + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$,$1$ |
| Indices of inseparability: | $[19, 14, 8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $128$ |
| Galois group: | $C_2^3:\OD_{16}$ (as 16T252) |
| Inertia group: | Intransitive group isomorphic to $C_2^3:C_4$ |
| Wild inertia group: | $C_2^3:C_4$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 3, \frac{7}{2}, 4]$ |
| Galois Swan slopes: | $[1,1,2,\frac{5}{2},3]$ |
| Galois mean slope: | $3.4375$ |
| Galois splitting model: |
$x^{16} - 88 x^{14} - 80 x^{13} + 938 x^{12} + 9080 x^{11} + 64344 x^{10} - 363400 x^{9} - 1007815 x^{8} + 5250800 x^{7} - 5290216 x^{6} - 7271200 x^{5} + 27011198 x^{4} - 14055520 x^{3} - 21743728 x^{2} + 24443320 x - 6459359$
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