Defining polynomial
|
\(x^{14} - 21574 x^{13} + 234933293 x^{12} - 1215548742590 x^{11} + 4112603302919993 x^{10} - 2725136947640868418 x^{9} + 363970304488058959670 x^{8} + 412439955621146008597774 x^{7} + 8371317003225356072410 x^{6} - 1441597445302019393122 x^{5} + 50038044386627554831 x^{4} - 340160375675128190 x^{3} + 1512111255867499 x^{2} - 3193726269286 x + 3404825447\)
|
Invariants
| Base field: | $\Q_{23}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $7$ |
| Discriminant exponent $c$: | $7$ |
| Discriminant root field: | $\Q_{23}(\sqrt{23\cdot 5})$ |
| Root number: | $-i$ |
| $\card{ \Gal(K/\Q_{ 23 }) }$: | $14$ |
| This field is Galois and abelian over $\Q_{23}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{23}(\sqrt{23\cdot 5})$, 23.7.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 23.7.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of
\( x^{7} + 21 x + 18 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + \left(184 t^{6} + 230 t^{5} + 161 t^{4} + 92 t^{3} + 460 t^{2} + 322 t + 230\right) x + 23 \)
$\ \in\Q_{23}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_{14}$ (as 14T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $7$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | Not computed |