Defining polynomial
|
$( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{3} + 4 x + 2$
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$ $=$$\Gal(K/\Q_{2})$: | $C_3:C_4$ |
| This field is Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2]$ |
| Visible Swan slopes: | $[1]$ |
| Means: | $\langle\frac{1}{2}\rangle$ |
| Rams: | $(3)$ |
| Jump set: | $[3, 12]$ |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.1.3.2a1.1 x3, 2.2.2.4a1.2, 2.2.3.4a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{6} + 2 x^{3} + 4 t + 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{4} + z^{2} + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
| Galois degree: | $12$ |
| Galois group: | $C_3:C_4$ (as 12T5) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Wild inertia group: | $C_2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[2]$ |
| Galois Swan slopes: | $[1]$ |
| Galois mean slope: | $1.3333333333333333$ |
| Galois splitting model: | $x^{12} - 39 x^{10} + 486 x^{8} - 120 x^{7} - 2259 x^{6} + 1380 x^{5} + 3126 x^{4} - 2980 x^{3} - 15 x^{2} + 420 x + 5$ |