Defining polynomial
|
$( x^{3} + x + 1 )^{7} + 2 x$
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $21$ |
| Ramification index $e$: | $7$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $18$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_7$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | $[7]$ |
| Roots of unity: | $14 = (2^{ 3 } - 1) \cdot 2$ |
Intermediate fields
| 2.3.1.0a1.1 |
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: |
\( x^{7} + 2 t \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^6 + z^5 + z^4 + z^3 + z^2 + z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $147$ |
| Galois group: | $C_7:C_{21}$ (as 21T13) |
| Inertia group: | Intransitive group isomorphic to $C_7$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $21$ |
| Galois tame degree: | $7$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.8571428571428571$ |
| Galois splitting model: | not computed |