Defining polynomial
\(x^{14} + 28 x^{13} + 350 x^{12} + 2576 x^{11} + 12404 x^{10} + 41104 x^{9} + 96152 x^{8} + 160650 x^{7} + 192444 x^{6} + 165676 x^{5} + 106232 x^{4} + 65016 x^{3} + 59920 x^{2} + 57232 x + 27193\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 5 }) }$: | $2$ |
This field is not Galois over $\Q_{5}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{5}(\sqrt{2})$, 5.7.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} + 4 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{7} + 5 \) $\ \in\Q_{5}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{6} + 2z^{5} + z^{4} + z + 2$ |
Associated inertia: | $3$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $F_7$ (as 14T4) |
Inertia group: | Intransitive group isomorphic to $C_7$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $6$ |
Tame degree: | $7$ |
Wild slopes: | None |
Galois mean slope: | $6/7$ |
Galois splitting model: | Not computed |