Defining polynomial
\(x^{8} - 1586 x^{4} - 198575\) |
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 13 }) }$: | $8$ |
This field is Galois and abelian over $\Q_{13}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{13}(\sqrt{2})$, 13.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified 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: | \( x^{4} + 156 t + 143 \) $\ \in\Q_{13}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{3} + 4z^{2} + 6z + 4$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_8$ (as 8T1) |
Inertia group: | Intransitive group isomorphic to $C_4$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $4$ |
Wild slopes: | None |
Galois mean slope: | $3/4$ |
Galois splitting model: | $x^{8} - x^{7} - 202 x^{6} + 235 x^{5} + 9269 x^{4} - 10074 x^{3} - 86808 x^{2} - 14600 x + 110800$ |