Defining polynomial
|
\(x^{10} - 3335 x^{9} + 21724972 x^{8} + 32175051716 x^{7} + 187094353927946 x^{6} + 562900511451090140 x^{5} + 3155504389465566 x^{4} - 101811190699272 x^{3} - 1679305412607 x^{2} - 10291712457 x - 115854174\)
|
Invariants
| Base field: | $\Q_{23}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $5$ |
| Discriminant root field: | $\Q_{23}(\sqrt{23})$ |
| Root number: | $-i$ |
| $\card{ \Gal(K/\Q_{ 23 }) }$: | $10$ |
| This field is Galois and abelian over $\Q_{23}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{23}(\sqrt{23})$, 23.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 23.5.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of
\( x^{5} + 3 x + 18 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + \left(345 t^{4} + 161 t^{3} + 253 t^{2} + 414 t + 161\right) x + 23 t \)
$\ \in\Q_{23}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_{10}$ (as 10T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $5$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | Not computed |