Normalized defining polynomial
\( x^{10} - x^{9} + 3x^{6} - 3x^{5} + 3x^{4} - 3x^{3} + 3x^{2} - x + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(0, 5)$ |
| |
| Discriminant: |
\(-5462760771\)
\(\medspace = -\,3^{9}\cdot 13\cdot 37\cdot 577\)
|
| |
| Root discriminant: | \(9.41\) |
| |
| Galois root discriminant: | $3^{9/10}13^{1/2}37^{1/2}577^{1/2}\approx 1416.0206084080282$ | ||
| Ramified primes: |
\(3\), \(13\), \(37\), \(577\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-832611}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$
| Monogenic: | Yes | |
| 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: |
\( a^{9} - a^{8} - a^{7} + 3 a^{5} - 3 a^{4} + a^{3} - 2 a^{2} + a \)
(order $6$)
|
| |
| Fundamental units: |
$a^{8}+2a^{4}-a^{3}+2a^{2}+1$, $a^{6}+a^{5}-a^{3}+a^{2}+a+1$, $a^{7}+a^{6}-a^{4}+2a^{3}+2a^{2}+2a$, $a^{9}-a^{8}-a^{7}+a^{6}+4a^{5}-3a^{4}-a^{2}+3a+1$
|
| |
| Regulator: | \( 16.2034361981 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 16.2034361981 \cdot 1}{6\cdot\sqrt{5462760771}}\cr\approx \mathstrut & 0.357807239853 \end{aligned}\]
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 | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 | $x^{10} + 3$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $$[\ ]_{10}^{4}$$ |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 13.1.2.1a1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 13.3.1.0a1.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 13.4.1.0a1.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 37.1.2.1a1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 37.2.1.0a1.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 37.2.1.0a1.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 37.3.1.0a1.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
|
\(577\)
| $\Q_{577}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |