Defining polynomial
\(x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $6$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 3 }) }$: | $12$ |
This field is Galois and abelian over $\Q_{3}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{3}(\sqrt{2})$, 3.3.0.1, 3.4.2.2, 3.6.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 3.6.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{2} + 3 t \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_{12}$ (as 12T1) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $6$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | $x^{12} - x^{11} - 22 x^{10} + 14 x^{9} + 153 x^{8} - 62 x^{7} - 396 x^{6} + 84 x^{5} + 361 x^{4} - 87 x^{3} - 112 x^{2} + 37 x + 1$ |