Defining polynomial
\(x^{12} - 12 x^{11} + 156 x^{10} - 1104 x^{9} + 3420 x^{8} + 972 x^{7} - 1890 x^{6} + 1404 x^{5} + 1944 x^{4} + 432 x^{3} + 324 x^{2} + 324 x + 81\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
Root number: | $-1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $2$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | $[3/2]$ |
Intermediate fields
$\Q_{3}(\sqrt{2})$, 3.4.0.1, 3.6.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 3.4.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{4} + 2 x^{3} + 2 \) |
Relative Eisenstein polynomial: | \( x^{3} + \left(3 t^{3} + 6 t^{2} + 6 t\right) x^{2} + \left(3 t^{3} + 3 t^{2} + 6\right) x + 3 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2t^{3} + 2t^{2} + 1$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2\times C_3^2:C_4$ (as 12T41) |
Inertia group: | Intransitive group isomorphic to $C_3:S_3$ |
Wild inertia group: | $C_3^2$ |
Unramified degree: | $4$ |
Tame degree: | $2$ |
Wild slopes: | $[3/2, 3/2]$ |
Galois mean slope: | $25/18$ |
Galois splitting model: | $x^{12} - 4 x^{9} - 6 x^{6} + 4 x^{3} + 1$ |