Defining polynomial
\(x^{12} - 6 x^{9} + 12 x^{6} + 18 x^{5} + 9 x^{4} + 36 x^{3} + 18 x^{2} + 18\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $14$ |
Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $4$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | $[3/2]$ |
Intermediate fields
$\Q_{3}(\sqrt{2})$, 3.3.3.1, 3.4.2.2, 3.6.6.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} + 2 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{6} + 3 t x^{3} + \left(3 t + 3\right) x^{2} + 3 t \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $2z^{2} + t$,$z^{3} + 2$ |
Associated inertia: | $2$,$1$ |
Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
Galois group: | $C_4\times S_3$ (as 12T11) |
Inertia group: | Intransitive group isomorphic to $S_3$ |
Wild inertia group: | $C_3$ |
Unramified degree: | $4$ |
Tame degree: | $2$ |
Wild slopes: | $[3/2]$ |
Galois mean slope: | $7/6$ |
Galois splitting model: | $x^{12} - 6 x^{11} + 15 x^{10} - 11 x^{9} - 27 x^{8} + 69 x^{7} - 40 x^{6} - 45 x^{5} + 57 x^{4} + 8 x^{3} - 24 x^{2} - 3 x + 1$ |