Defining polynomial
|
\(x^{14} + 168 x^{13} + 12110 x^{12} + 485856 x^{11} + 11733204 x^{10} + 171095904 x^{9} + 1407877912 x^{8} + 5266970938 x^{7} + 2815760696 x^{6} + 684731964 x^{5} + 107750832 x^{4} + 339857336 x^{3} + 4765729696 x^{2} + 37989914704 x + 129797258121\)
|
Invariants
| Base field: | $\Q_{29}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{29}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 29 }) }$: | $14$ |
| This field is Galois and abelian over $\Q_{29}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{29}(\sqrt{2})$, 29.7.6.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{29}(\sqrt{2})$ $\cong \Q_{29}(t)$ where $t$ is a root of
\( x^{2} + 24 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{7} + 29 \)
$\ \in\Q_{29}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{6} + 7z^{5} + 21z^{4} + 6z^{3} + 6z^{2} + 21z + 7$ |
| 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 |