Defining polynomial
\(x^{10} + 2 x^{9} + 2 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x + 2\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $10$ |
Ramification exponent $e$: | $10$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $18$ |
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: | $[14/5]$ |
Intermediate fields
2.5.4.1 |
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: | \( x^{10} + 2 x^{9} + 2 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x + 2 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{8} + z^{6} + 1$ |
Associated inertia: | $1$,$4$ |
Indices of inseparability: | $[9, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2^4:F_5$ (as 10T25) |
Inertia group: | $C_2^4:C_5$ (as 10T8) |
Wild inertia group: | $C_2^4$ |
Unramified degree: | $4$ |
Tame degree: | $5$ |
Wild slopes: | $[14/5, 14/5, 14/5, 14/5]$ |
Galois mean slope: | $107/40$ |
Galois splitting model: | $x^{10} + 5 x^{8} - 2 x^{6} - 32 x^{4} + 55 x^{2} - 13$ |