Defining polynomial
\(x^{8} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 30\)
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Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $8$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $24$ |
Discriminant root field: | $\Q_{2}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 2 }) }$: | $8$ |
This field is Galois over $\Q_{2}.$ | |
Visible slopes: | $[2, 3, 4]$ |
Intermediate fields
$\Q_{2}(\sqrt{-5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, $\Q_{2}(\sqrt{2})$, 2.4.8.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}$ |
Relative Eisenstein polynomial: |
\( x^{8} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 30 \)
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Ramification polygon
Not computedInvariants of the Galois closure
Galois group: | $Q_8$ (as 8T5) |
Inertia group: | $Q_8$ (as 8T5) |
Wild inertia group: | $Q_8$ |
Unramified degree: | $1$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 3, 4]$ |
Galois mean slope: | $3$ |
Galois splitting model: | $x^{8} - 12 x^{6} + 36 x^{4} - 36 x^{2} + 9$ |