Defining polynomial
\(x^{8} + 14 x^{4} - 245\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{7}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 7 }) }$: | $8$ |
This field is Galois over $\Q_{7}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{7}(\sqrt{3})$, $\Q_{7}(\sqrt{7})$, $\Q_{7}(\sqrt{7\cdot 3})$, 7.4.2.1 |
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: | \( x^{4} + 7 t + 28 \) $\ \in\Q_{7}(t)[x]$ |
Ramification polygon
Data not computedInvariants of the Galois closure
Galois group: | $Q_8$ (as 8T5) |
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: | $x^{8} - 3 x^{7} + 22 x^{6} - 60 x^{5} + 201 x^{4} - 450 x^{3} + 1528 x^{2} - 3069 x + 4561$ |