Normalized defining polynomial
\( x^{10} - 4x^{9} + 9x^{8} - 12x^{7} + 12x^{6} - 9x^{5} + 9x^{4} - 9x^{3} + 9x^{2} - 4x + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(0, 5)$ |
| |
| Discriminant: |
\(-3219371163\)
\(\medspace = -\,3^{9}\cdot 163561\)
|
| |
| Root discriminant: | \(8.93\) |
| |
| Galois root discriminant: | $3^{9/10}163561^{1/2}\approx 1087.0487170534657$ | ||
| Ramified primes: |
\(3\), \(163561\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-490683}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{2}{5}a^{8}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{2}{5}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -\frac{3}{5} a^{9} + \frac{11}{5} a^{8} - 5 a^{7} + \frac{31}{5} a^{6} - \frac{29}{5} a^{5} + \frac{19}{5} a^{4} - \frac{24}{5} a^{3} + \frac{24}{5} a^{2} - \frac{24}{5} a + \frac{9}{5} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+a^{7}-\frac{7}{5}a^{6}+\frac{8}{5}a^{5}-\frac{3}{5}a^{4}+\frac{3}{5}a^{3}-\frac{3}{5}a^{2}+\frac{8}{5}a-\frac{3}{5}$, $\frac{4}{5}a^{9}-\frac{13}{5}a^{8}+5a^{7}-\frac{28}{5}a^{6}+\frac{27}{5}a^{5}-\frac{22}{5}a^{4}+\frac{27}{5}a^{3}-\frac{27}{5}a^{2}+\frac{22}{5}a-\frac{7}{5}$, $\frac{3}{5}a^{9}-\frac{16}{5}a^{8}+7a^{7}-\frac{46}{5}a^{6}+\frac{39}{5}a^{5}-\frac{29}{5}a^{4}+\frac{24}{5}a^{3}-\frac{34}{5}a^{2}+\frac{29}{5}a-\frac{9}{5}$, $\frac{3}{5}a^{9}-\frac{11}{5}a^{8}+5a^{7}-\frac{31}{5}a^{6}+\frac{29}{5}a^{5}-\frac{19}{5}a^{4}+\frac{24}{5}a^{3}-\frac{19}{5}a^{2}+\frac{19}{5}a-\frac{4}{5}$
|
| |
| Regulator: | \( 11.2626362495 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 11.2626362495 \cdot 1}{6\cdot\sqrt{3219371163}}\cr\approx \mathstrut & 0.323968380204 \end{aligned}\]
Galois group
$S_5\wr C_2$ (as 10T43):
| A non-solvable group of order 28800 |
| The 35 conjugacy class representatives for $S_5^2 \wr C_2$ |
| Character table for $S_5^2 \wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 25 sibling: | data not computed |
| Degree 30 sibling: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.10.9a1.1 | $x^{10} + 3$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $$[\ ]_{10}^{4}$$ |
|
\(163561\)
| $\Q_{163561}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |