Defining polynomial
\(x^{12} - 21970 x^{6} + 314171 x^{4} - 4084223 x^{2} + 9653618\) |
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $6$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 13 }) }$: | $12$ |
This field is Galois and abelian over $\Q_{13}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{13}(\sqrt{2})$, 13.3.0.1, 13.4.2.2, 13.6.0.1 |
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 \( x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{2} + 13 t \) $\ \in\Q_{13}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |