Defining polynomial
\(x^{6} + x^{2} - 2 x + 2\) ![]() |
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $6$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
Root number: | $1$ |
$|\Gal(K/\Q_{ 13 })|$: | $6$ |
This field is Galois and abelian over $\Q_{13}.$ |
Intermediate fields
$\Q_{13}(\sqrt{2})$, 13.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 13.6.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{6} + x^{2} - 2 x + 2 \) ![]() |
Relative Eisenstein polynomial: | \( x - 13 \)$\ \in\Q_{13}(t)[x]$ ![]() |
Invariants of the Galois closure
Galois group: | $C_6$ (as 6T1) |
Inertia group: | trivial |
Unramified degree: | $6$ |
Tame degree: | $1$ |
Wild slopes: | None |
Galois mean slope: | $0$ |
Galois splitting model: | $x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$ |