Defining polynomial
\(x^{14} + 294 x^{13} + 37065 x^{12} + 2598372 x^{11} + 109465209 x^{10} + 2775672522 x^{9} + 39406741353 x^{8} + 247146613646 x^{7} + 118220236701 x^{6} + 24982640172 x^{5} + 3066494193 x^{4} + 4861739862 x^{3} + 117021374001 x^{2} + 1635382592172 x + 9795578218654\) |
Invariants
Base field: | $\Q_{43}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{43}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 43 }) }$: | $14$ |
This field is Galois and abelian over $\Q_{43}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{43}(\sqrt{2})$, 43.7.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{43}(\sqrt{2})$ $\cong \Q_{43}(t)$ where $t$ is a root of \( x^{2} + 42 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{7} + 43 \) $\ \in\Q_{43}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{6} + 7z^{5} + 21z^{4} + 35z^{3} + 35z^{2} + 21z + 7$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_{14}$ (as 14T1) |
Inertia group: | Intransitive group isomorphic to $C_7$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $7$ |
Wild slopes: | None |
Galois mean slope: | $6/7$ |
Galois splitting model: | Not computed |