Defining polynomial
\(x^{12} + 24 x^{11} + 216 x^{10} + 768 x^{9} - 432 x^{8} - 10368 x^{7} - 18414 x^{6} + 27864 x^{5} + 83592 x^{4} + 10800 x^{3} + 64800 x^{2} + 901125\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $16$ |
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: | $[2]$ |
Intermediate fields
$\Q_{3}(\sqrt{2})$, 3.3.4.2, 3.4.0.1, 3.6.8.3 |
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: | \( x^{3} + 6 x^{2} + 9 t^{3} + 18 t + 3 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
Galois group: | $C_{12}$ (as 12T1) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Wild inertia group: | $C_3$ |
Unramified degree: | $4$ |
Tame degree: | $1$ |
Wild slopes: | $[2]$ |
Galois mean slope: | $4/3$ |
Galois splitting model: | $x^{12} + 3 x^{10} - x^{9} + 9 x^{8} + 9 x^{7} + 28 x^{6} + 18 x^{5} + 75 x^{4} + 26 x^{3} + 9 x^{2} + 3 x + 1$ |