Defining polynomial
|
$( x^{3} + x + 1 )^{6} + 2 x^{2} ( x^{3} + x + 1 )^{2} + \left(2 x + 2\right) ( x^{3} + x + 1 ) + 2$
|
Invariants
| Base field: | $\Q_{2}$ |
|
| Degree $d$: | $18$ |
|
| Ramification index $e$: | $6$ |
|
| Residue field degree $f$: | $3$ |
|
| Discriminant exponent $c$: | $18$ |
|
| Discriminant root field: | $\Q_{2}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{2})$: | $C_2$ | |
| This field is not Galois over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[\frac{4}{3}]$ | |
| Visible Swan slopes: | $[\frac{1}{3}]$ | |
| Means: | $\langle\frac{1}{6}\rangle$ | |
| Rams: | $(1)$ | |
| Jump set: | $[3, 7]$ | |
| Roots of unity: | $14 = (2^{ 3 } - 1) \cdot 2$ |
|
Intermediate fields
| 2.3.1.0a1.1, 2.1.3.2a1.1, 2.3.3.6a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.3.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{3} + x + 1 \)
|
|
| Relative Eisenstein polynomial: |
\( x^{6} + 2 t^{2} x^{2} + \left(2 t + 2\right) x + 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^4 + z^2 + 1$,$z + t^2$ |
| Associated inertia: | $2$,$1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $288$ |
| Galois group: | $A_4\wr C_2$ (as 18T113) |
| Inertia group: | Intransitive group isomorphic to $C_2^2:A_4$ |
| Wild inertia group: | $C_2^4$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}]$ |
| Galois mean slope: | $1.2916666666666667$ |
| Galois splitting model: |
$x^{18} - 2 x^{17} - 8 x^{16} + 12 x^{15} + 19 x^{14} - 16 x^{13} - 23 x^{12} - 18 x^{11} + 119 x^{10} + 140 x^{9} - 224 x^{8} - 246 x^{7} + 42 x^{6} + 76 x^{5} + 39 x^{4} + 64 x^{3} + 55 x^{2} + 18 x + 1$
|