Defining polynomial
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\(x^{8} + 10 x^{6} + 10 x^{4} + 8 x^{3} + 8 x^{2} + 8 x + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $21$ |
| Discriminant root field: | $\Q_{2}(\sqrt{2\cdot 5})$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2, 2, 15/4]$ |
Intermediate fields
| 2.4.6.7 |
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: |
\( x^{8} + 10 x^{6} + 10 x^{4} + 8 x^{3} + 8 x^{2} + 8 x + 2 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{6} + z^{2} + 1$ |
| Associated inertia: | $1$,$3$ |
| Indices of inseparability: | $[14, 6, 4, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\wr A_4$ (as 8T38) |
| Inertia group: | $C_2\wr C_2^2$ (as 8T31) |
| Wild inertia group: | $C_2\wr C_2^2$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 3, 7/2, 7/2, 15/4]$ |
| Galois mean slope: | $111/32$ |
| Galois splitting model: | $x^{8} - 4 x^{6} + 40 x^{4} - 64 x^{2} + 416$ |