Defining polynomial
\(x^{8} + 8 x^{7} + 60 x^{6} + 136 x^{5} + 256 x^{4} + 240 x^{3} + 104 x^{2} + 112 x + 76\)
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Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $22$ |
Discriminant root field: | $\Q_{2}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 2 }) }$: | $8$ |
This field is Galois over $\Q_{2}.$ | |
Visible slopes: | $[3, 4]$ |
Intermediate fields
$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.4.6.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified 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: |
\( x^{4} + \left(8 t + 8\right) x^{3} + \left(12 t + 4\right) x^{2} + \left(8 t + 8\right) x + 4 t + 10 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
Not computedInvariants of the Galois closure
Galois group: | $Q_8$ (as 8T5) |
Inertia group: | Intransitive group isomorphic to $C_4$ |
Wild inertia group: | $C_4$ |
Unramified degree: | $2$ |
Tame degree: | $1$ |
Wild slopes: | $[3, 4]$ |
Galois mean slope: | $11/4$ |
Galois splitting model: | $x^{8} - 60 x^{6} + 810 x^{4} - 1800 x^{2} + 900$ |