Defining polynomial
\(x^{12} + 6 x^{10} + 6 x^{6} + 9 x^{3} + 9 x^{2} + 9 x + 24\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $12$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $21$ |
Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $3$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | $[9/4]$ |
Intermediate fields
$\Q_{3}(\sqrt{3})$, 3.4.3.2 |
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: | \( x^{12} + 6 x^{10} + 6 x^{6} + 9 x^{3} + 9 x^{2} + 9 x + 24 \) |
Ramification polygon
Residual polynomials: | $z^{2} + 2$,$z^{9} + z^{6} + 1$ |
Associated inertia: | $1$,$2$ |
Indices of inseparability: | $[10, 0]$ |
Invariants of the Galois closure
Galois group: | $C_3\wr D_4$ (as 12T167) |
Inertia group: | $C_3^3:C_4$ (as 12T72) |
Wild inertia group: | $C_3^3$ |
Unramified degree: | $6$ |
Tame degree: | $4$ |
Wild slopes: | $[3/2, 9/4, 9/4]$ |
Galois mean slope: | $77/36$ |
Galois splitting model: | $x^{12} - 42 x^{10} - 14 x^{9} - 378 x^{8} + 504 x^{7} + 5334 x^{6} - 5292 x^{5} + 109368 x^{4} - 73892 x^{3} + 31752 x^{2} - 1176 x - 980$ |