Defining polynomial
\(x^{8} - 154 x^{4} - 1421\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
Root number: | $-1$ |
$\card{ \Aut(K/\Q_{ 7 }) }$: | $4$ |
This field is not Galois over $\Q_{7}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{7}(\sqrt{3})$, 7.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} + 6 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{4} + 35 t + 28 \) $\ \in\Q_{7}(t)[x]$ |
Ramification polygon
Data not computedInvariants of the Galois closure
Galois group: | $\OD_{16}$ (as 8T7) |
Inertia group: | Intransitive group isomorphic to $C_4$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $4$ |
Wild slopes: | None |
Galois mean slope: | $3/4$ |
Galois splitting model: | $x^{8} - x^{7} - 13 x^{6} - 13 x^{5} + 25 x^{4} + 38 x^{3} - 33 x^{2} - 34 x + 11$ |