Defining polynomial
|
$( x^{4} + 3 x^{2} + 12 x + 2 )^{2} + 13$
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $8$ |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{13}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{13})$ $=$$\Gal(K/\Q_{13})$: | $C_2\times C_4$ |
| This field is Galois and abelian over $\Q_{13}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $28560 = (13^{ 4 } - 1)$ |
Intermediate fields
| $\Q_{13}(\sqrt{2})$, $\Q_{13}(\sqrt{13})$, $\Q_{13}(\sqrt{13\cdot 2})$, 13.4.1.0a1.1, 13.2.2.2a1.2, 13.2.2.2a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 13.4.1.0a1.1 $\cong \Q_{13}(t)$ where $t$ is a root of
\( x^{4} + 3 x^{2} + 12 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 13 \)
$\ \in\Q_{13}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $8$ |
| Galois group: | $C_2\times C_4$ (as 8T2) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[]$ |
| Galois mean slope: | $0.5$ |
| Galois splitting model: | $x^{8} - x^{7} + 4 x^{6} - 7 x^{5} + 19 x^{4} + 21 x^{3} + 36 x^{2} + 27 x + 81$ |