Defining polynomial
\(x^{8} - 16 x^{6} + 112 x^{5} + 56 x^{4} - 1280 x^{3} + 10176 x^{2} + 39744 x + 40720\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[3]$ |
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: | \( x^{2} + 4 t x + 8 t^{3} + 4 t^{2} + 12 t + 2 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 1$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
Galois group: | $\OD_{16}:C_2$ (as 8T16) |
Inertia group: | Intransitive group isomorphic to $C_2^3$ |
Wild inertia group: | $C_2^3$ |
Unramified degree: | $4$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 2, 3]$ |
Galois mean slope: | $9/4$ |
Galois splitting model: | $x^{8} - 10 x^{6} + 40 x^{4} - 80 x^{2} + 80$ |