Defining polynomial
\(x^{8} + 8 x^{7} + 36 x^{6} + 80 x^{5} + 104 x^{4} - 32 x^{3} - 8 x^{2} + 124\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $20$ |
Discriminant root field: | $\Q_{2}$ |
Root number: | $-1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $4$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[3, 7/2]$ |
Intermediate fields
$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{2})$, $\Q_{2}(\sqrt{2\cdot 5})$, 2.4.6.1, 2.4.10.3, 2.4.10.2 |
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: | \( x^{4} + 4 x^{3} + \left(4 t + 12\right) x^{2} + 12 t + 2 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{2} + 1$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[7, 4, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2\times D_4$ (as 8T9) |
Inertia group: | Intransitive group isomorphic to $D_4$ |
Wild inertia group: | $D_4$ |
Unramified degree: | $2$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 3, 7/2]$ |
Galois mean slope: | $11/4$ |
Galois splitting model: | $x^{8} - 2 x^{6} + 5 x^{4} + 2 x^{2} + 1$ |