Defining polynomial
\(x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $2$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{3}(\sqrt{2})$, 3.7.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} + 2 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{7} + 3 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{6} + z^{5} + 2z^{3} + 2z^{2} + 1$ |
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 |