Defining polynomial
\(x^{8} + 4 x^{7} + 4 x^{6} + 12\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{2}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $4$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[2, 2]$ |
Intermediate fields
$\Q_{2}(\sqrt{5})$, 2.4.4.2, 2.4.4.3, 2.4.4.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} + x + 1 \) |
Relative Eisenstein polynomial: | \( x^{4} + 2 x^{3} + 4 t + 2 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{3} + 1$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[3, 3, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2^2:C_4$ (as 8T10) |
Inertia group: | Intransitive group isomorphic to $C_2^2$ |
Wild inertia group: | $C_2^2$ |
Unramified degree: | $4$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 2]$ |
Galois mean slope: | $3/2$ |
Galois splitting model: | $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 14 x^{4} - 8 x^{3} - 16 x^{2} + 16 x - 4$ |