Defining polynomial
\(x^{6} + 2 x + 2\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $6$ |
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: | $[4/3]$ |
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: | \( x^{6} + 2 x + 2 \) |
Ramification polygon
Data not computedInvariants of the Galois closure
Galois group: | $S_4$ (as 6T8) |
Inertia group: | $A_4$ (as 6T4) |
Wild inertia group: | $C_2^2$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | $[4/3, 4/3]$ |
Galois mean slope: | $7/6$ |
Galois splitting model: | $x^{6} - 12 x^{3} + 18 x + 30$ |