Defining polynomial
\(x^{15} + 2 x^{7} + 12 x^{6} + 2 x^{5} + 11 x^{4} + 10 x^{3} + 11 x^{2} + 8 x + 11\) |
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $15$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{13}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 13 }) }$: | $15$ |
This field is Galois and abelian over $\Q_{13}.$ | |
Visible slopes: | None |
Intermediate fields
13.3.0.1, 13.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 13.15.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{15} + 2 x^{7} + 12 x^{6} + 2 x^{5} + 11 x^{4} + 10 x^{3} + 11 x^{2} + 8 x + 11 \) |
Relative Eisenstein polynomial: | \( x - 13 \) $\ \in\Q_{13}(t)[x]$ |
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
Galois group: | $C_{15}$ (as 15T1) |
Inertia group: | trivial |
Wild inertia group: | $C_1$ |
Unramified degree: | $15$ |
Tame degree: | $1$ |
Wild slopes: | None |
Galois mean slope: | $0$ |
Galois splitting model: | Not computed |