Defining polynomial
\(x^{12} - 6 x^{11} + 57 x^{10} - 42 x^{9} + 126 x^{8} - 36 x^{7} + 51 x^{6} - 18 x^{5} + 36 x^{4} - 18 x^{3} + 9\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $18$ |
Discriminant root field: | $\Q_{3}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $6$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
$\Q_{3}(\sqrt{2})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3\cdot 2})$, 3.4.2.1, 3.6.9.14 |
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: | \( x^{6} + \left(6 t + 3\right) x^{5} + 6 x^{4} + \left(6 t + 3\right) x^{3} + 3 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $2z^{2} + 1$,$z^{3} + 2$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
Galois group: | $C_6\times S_3$ (as 12T18) |
Inertia group: | Intransitive group isomorphic to $C_3\times S_3$ |
Wild inertia group: | $C_3^2$ |
Unramified degree: | $2$ |
Tame degree: | $2$ |
Wild slopes: | $[3/2, 2]$ |
Galois mean slope: | $31/18$ |
Galois splitting model: | $x^{12} - 8 x^{9} + 23 x^{6} + 14 x^{3} + 7$ |