Defining polynomial
|
$( x^{2} + 2 x + 2 )^{6} + 6 ( x^{2} + 2 x + 2 ) + 3 x$
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{3})$: | $C_1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[\frac{5}{4}]$ |
| Visible Swan slopes: | $[\frac{1}{4}]$ |
| Means: | $\langle\frac{1}{6}\rangle$ |
| Rams: | $(\frac{1}{2})$ |
| Jump set: | undefined |
| Roots of unity: | $8 = (3^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.2.2.2a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical 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: |
\( x^{6} + 3 t x + 3 t \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{3} + 2$,$2 z + (t + 1)$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $1296$ |
| Galois group: | $C_3^4:\OD_{16}$ (as 12T215) |
| Inertia group: | Intransitive group isomorphic to $C_3^4:C_4$ |
| Wild inertia group: | $C_3^4$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{5}{4}, \frac{5}{4}, \frac{5}{4}, \frac{5}{4}]$ |
| Galois Swan slopes: | $[\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}]$ |
| Galois mean slope: | $1.2438271604938271$ |
| Galois splitting model: |
$x^{12} - 6 x^{11} + 12 x^{10} - 3 x^{9} - 15 x^{8} + 18 x^{7} - 13 x^{6} + 6 x^{5} + 21 x^{4} - 45 x^{3} + 15 x^{2} + 15 x - 5$
|