Defining polynomial
\(x^{8} + 130 x^{6} + 32 x^{5} + 5967 x^{4} - 1888 x^{3} + 113114 x^{2} - 89184 x + 766577\) |
Invariants
Base field: | $\Q_{31}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $4$ |
Discriminant root field: | $\Q_{31}$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 31 }) }$: | $8$ |
This field is Galois and abelian over $\Q_{31}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{31}(\sqrt{3})$, $\Q_{31}(\sqrt{31})$, $\Q_{31}(\sqrt{31\cdot 3})$, 31.4.0.1, 31.4.2.1, 31.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 31.4.0.1 $\cong \Q_{31}(t)$ where $t$ is a root of \( x^{4} + 3 x^{2} + 16 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{2} + 31 \) $\ \in\Q_{31}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_2\times C_4$ (as 8T2) |
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 |