Defining polynomial
\(x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 10\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $11$ |
Discriminant root field: | $\Q_{2}(\sqrt{2\cdot 5})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[3]$ |
Intermediate fields
2.3.2.1 |
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: | \( x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 10 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{4} + z^{2} + 1$ |
Associated inertia: | $1$,$2$ |
Indices of inseparability: | $[6, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2\times S_4$ (as 6T11) |
Inertia group: | $C_2\times A_4$ (as 6T6) |
Wild inertia group: | $C_2^3$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | $[4/3, 4/3, 3]$ |
Galois mean slope: | $25/12$ |
Galois splitting model: | $x^{6} + 6 x^{4} + 6$ |