Defining polynomial
\(x^{6} + x^{4} + 10 x^{3} + 13 x^{2} + 29 x + 7\) |
Invariants
Base field: | $\Q_{71}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $6$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{71}(\sqrt{7})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 71 }) }$: | $6$ |
This field is Galois and abelian over $\Q_{71}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{71}(\sqrt{7})$, 71.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 71.6.0.1 $\cong \Q_{71}(t)$ where $t$ is a root of \( x^{6} + x^{4} + 10 x^{3} + 13 x^{2} + 29 x + 7 \) |
Relative Eisenstein polynomial: | \( x - 71 \) $\ \in\Q_{71}(t)[x]$ |
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
Galois group: | $C_6$ (as 6T1) |
Inertia group: | trivial |
Wild inertia group: | $C_1$ |
Unramified degree: | $6$ |
Tame degree: | $1$ |
Wild slopes: | None |
Galois mean slope: | $0$ |
Galois splitting model: | $x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1$ |