Defining polynomial
|
$( x^{2} + 12 x + 2 )^{7} + 130 x + 26$
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $14$ |
| Ramification index $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{13})$: | $C_7$ |
| This field is not Galois over $\Q_{13}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $168 = (13^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{13}(\sqrt{2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{13}(\sqrt{2})$ $\cong \Q_{13}(t)$ where $t$ is a root of
\( x^{2} + 12 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{7} + 156 t + 143 \)
$\ \in\Q_{13}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{6} + 7 z^{5} + 8 z^{4} + 9 z^{3} + 9 z^{2} + 8 z + 7$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $98$ |
| Galois group: | $C_7\times D_7$ (as 14T8) |
| Inertia group: | Intransitive group isomorphic to $C_7$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $14$ |
| Galois tame degree: | $7$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[]$ |
| Galois mean slope: | $0.8571428571428571$ |
| Galois splitting model: | not computed |