Defining polynomial
\(x^{8} + 722 x^{4} - 75449 x^{2} + 260642\) |
Invariants
Base field: | $\Q_{19}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $4$ |
Discriminant root field: | $\Q_{19}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 19 }) }$: | $8$ |
This field is Galois and abelian over $\Q_{19}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{19}(\sqrt{2})$, 19.4.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 19.4.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{4} + 2 x^{2} + 11 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{2} + 19 t \) $\ \in\Q_{19}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_8$ (as 8T1) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | Not computed |