Defining polynomial
\(x^{6} + 2 x^{3} + 2\)
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Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
$\Q_{2}(\sqrt{-1})$, 2.3.2.1 |
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: |
\( x^{6} + 2 x^{3} + 2 \)
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Ramification polygon
Residual polynomials: | $z + 1$,$z^{4} + z^{2} + 1$ |
Associated inertia: | $1$,$2$ |
Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
Galois group: | $D_6$ (as 6T3) |
Inertia group: | $C_6$ (as 6T1) |
Wild inertia group: | $C_2$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | $[2]$ |
Galois mean slope: | $4/3$ |
Galois splitting model: |
$x^{6} + 2 x^{3} + 2$
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