Defining polynomial
\(x^{12} + 36 x^{11} + 540 x^{10} + 3900 x^{9} + 8028 x^{8} - 44388 x^{7} + 21870 x^{6} + 7020 x^{5} + 20412 x^{4} - 8316 x^{3} + 4212 x^{2} - 972 x + 81\) |
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: | $-1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | $[3/2]$ |
Intermediate fields
$\Q_{3}(\sqrt{2})$, 3.4.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 3.4.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{4} + 2 x^{3} + 2 \) |
Relative Eisenstein polynomial: | \( x^{3} + \left(6 t^{2} + 6 t + 6\right) x^{2} + \left(3 t^{3} + 6 t\right) x + 3 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2t^{3} + t$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois group: | $C_3^3:(C_4\times S_3)$ (as 12T170) |
Inertia group: | Intransitive group isomorphic to $C_3^3:S_3$ |
Wild inertia group: | $C_3^4$ |
Unramified degree: | $4$ |
Tame degree: | $2$ |
Wild slopes: | $[3/2, 3/2, 3/2, 3/2]$ |
Galois mean slope: | $241/162$ |
Galois splitting model: | $x^{12} - 8 x^{9} + 16 x^{6} - 8 x^{3} - 2$ |