Defining polynomial
\(x^{8} - 6 x^{7} + 40 x^{6} - 52 x^{5} + 192 x^{4} + 328 x^{3} + 1376 x^{2} + 1264 x + 1328\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $4$ |
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
$\Q_{2}(\sqrt{5})$, 2.4.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 2.4.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{4} + x + 1 \) |
Relative Eisenstein polynomial: | \( x^{2} + \left(2 t^{3} + 2 t\right) x + 4 t^{3} + 4 t + 6 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + t^{3} + t$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2\wr C_4$ (as 8T27) |
Inertia group: | Intransitive group isomorphic to $C_2^4$ |
Wild inertia group: | $C_2^4$ |
Unramified degree: | $4$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 2, 2, 2]$ |
Galois mean slope: | $15/8$ |
Galois splitting model: | $x^{8} - 6 x^{6} + 6 x^{4} + 9 x^{2} - 9$ |