Normalized defining polynomial
\( x^{10} - 3x^{9} + 4x^{8} - 4x^{7} + 4x^{6} - 5x^{4} + 4x^{3} - x + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $[0, 5]$ |
| |
| Discriminant: |
\(-443229763\)
\(\medspace = -\,17^{2}\cdot 97^{2}\cdot 163\)
|
| |
| Root discriminant: | \(7.32\) |
| |
| Galois root discriminant: | $17^{1/2}97^{1/2}163^{1/2}\approx 518.446718573857$ | ||
| Ramified primes: |
\(17\), \(97\), \(163\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-163}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$
| 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: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{2}{3}a^{9}-\frac{5}{3}a^{8}+\frac{7}{3}a^{7}-3a^{6}+\frac{11}{3}a^{5}-\frac{5}{3}a^{4}-\frac{2}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{2}{3}$, $a$, $\frac{1}{3}a^{9}-\frac{4}{3}a^{8}+\frac{5}{3}a^{7}-2a^{6}+\frac{7}{3}a^{5}-\frac{4}{3}a^{4}-\frac{4}{3}a^{3}+\frac{2}{3}a^{2}+\frac{1}{3}a-\frac{2}{3}$, $2a^{9}-5a^{8}+6a^{7}-7a^{6}+7a^{5}-6a^{3}+3a^{2}-1$
|
| |
| Regulator: | \( 0.943232786232 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 0.943232786232 \cdot 1}{2\cdot\sqrt{443229763}}\cr\approx \mathstrut & 0.2193682529095 \end{aligned}\]
Galois group
$C_2\wr S_5$ (as 10T39):
| A non-solvable group of order 3840 |
| The 36 conjugacy class representatives for $C_2 \wr S_5$ |
| Character table for $C_2 \wr S_5$ |
Intermediate fields
| 5.1.1649.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 32 siblings: | 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 | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.2.2.2a1.2 | $x^{4} + 32 x^{3} + 262 x^{2} + 96 x + 26$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 17.6.1.0a1.1 | $x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | |
|
\(97\)
| $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 97.2.1.0a1.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 97.2.1.0a1.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 97.2.2.2a1.2 | $x^{4} + 192 x^{3} + 9226 x^{2} + 960 x + 122$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(163\)
| 163.1.2.1a1.1 | $x^{2} + 163$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 163.4.1.0a1.1 | $x^{4} + 8 x^{2} + 91 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 163.4.1.0a1.1 | $x^{4} + 8 x^{2} + 91 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |