Defining polynomial
\(x^{8} + 16 x^{3} + 16 x^{2} + 16 x + 58\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $8$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $31$ |
Discriminant root field: | $\Q_{2}(\sqrt{-2})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[3, 4, 5]$ |
Intermediate fields
$\Q_{2}(\sqrt{-2\cdot 5})$, 2.4.11.12 |
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: | \( x^{8} + 16 x^{3} + 16 x^{2} + 16 x + 58 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{4} + 1$ |
Associated inertia: | $1$,$1$,$1$ |
Indices of inseparability: | $[24, 16, 8, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2\wr D_4$ (as 8T35) |
Inertia group: | $C_2\wr C_4$ (as 8T28) |
Wild inertia group: | $C_2\wr C_4$ |
Unramified degree: | $2$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 3, 7/2, 4, 17/4, 5]$ |
Galois mean slope: | $141/32$ |
Galois splitting model: | $x^{8} - 8 x^{6} - 164 x^{4} + 1800 x^{2} - 4050$ |