Defining polynomial
\(x^{6} + 3 x^{5} + 9 x^{2} + 9 x + 24\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | $[9/4]$ |
Intermediate fields
$\Q_{3}(\sqrt{3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{3}$ |
Relative Eisenstein polynomial: | \( x^{6} + 3 x^{5} + 9 x^{2} + 9 x + 24 \) |
Ramification polygon
Residual polynomials: | $2z + 2$,$z^{3} + 2$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[5, 0]$ |
Invariants of the Galois closure
Galois group: | $\SOPlus(4,2)$ (as 6T13) |
Inertia group: | $C_3^2:C_4$ (as 6T10) |
Wild inertia group: | $C_3^2$ |
Unramified degree: | $2$ |
Tame degree: | $4$ |
Wild slopes: | $[9/4, 9/4]$ |
Galois mean slope: | $25/12$ |
Galois splitting model: | $x^{6} - 6 x^{4} - 2 x^{3} + 9 x^{2} + 6 x - 2$ |