Normalized defining polynomial
\( x^{10} - x^{9} - x^{8} + 3x^{6} - x^{5} + 4x^{3} + 2x^{2} + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(0, 5)$ |
| |
| Discriminant: |
\(-6807776192\)
\(\medspace = -\,2^{6}\cdot 7^{5}\cdot 6329\)
|
| |
| Root discriminant: | \(9.62\) |
| |
| Galois root discriminant: | $2^{3/2}7^{1/2}6329^{1/2}\approx 595.3351996984555$ | ||
| Ramified primes: |
\(2\), \(7\), \(6329\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-44303}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-7}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{134}a^{9}-\frac{21}{67}a^{8}-\frac{21}{134}a^{7}+\frac{57}{134}a^{6}-\frac{28}{67}a^{5}+\frac{17}{134}a^{4}-\frac{27}{134}a^{3}+\frac{39}{134}a^{2}+\frac{11}{134}a-\frac{49}{134}$
| 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: |
$a$, $\frac{30}{67}a^{9}-\frac{54}{67}a^{8}-\frac{27}{67}a^{7}+\frac{35}{67}a^{6}+\frac{129}{67}a^{5}-\frac{93}{67}a^{4}-\frac{73}{67}a^{3}+\frac{98}{67}a^{2}-\frac{5}{67}a-\frac{63}{67}$, $\frac{31}{67}a^{9}-\frac{29}{67}a^{8}-\frac{48}{67}a^{7}+\frac{25}{67}a^{6}+\frac{73}{67}a^{5}-\frac{9}{67}a^{4}-\frac{33}{67}a^{3}+\frac{137}{67}a^{2}+\frac{73}{67}a-\frac{45}{67}$, $\frac{24}{67}a^{9}-\frac{3}{67}a^{8}-\frac{35}{67}a^{7}-\frac{39}{67}a^{6}+\frac{63}{67}a^{5}+\frac{73}{67}a^{4}+\frac{22}{67}a^{3}+\frac{65}{67}a^{2}+\frac{130}{67}a+\frac{97}{67}$
|
| |
| Regulator: | \( 12.573797171 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 12.573797171 \cdot 1}{2\cdot\sqrt{6807776192}}\cr\approx \mathstrut & 0.74616205817 \end{aligned}\]
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
| \(\Q(\sqrt{-7}) \) |
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 | R | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.10.0.1}{10} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 2.1.4.6a2.1 | $x^{4} + 2 x^{3} + 2 x^{2} + 2$ | $4$ | $1$ | $6$ | $A_4$ | $$[2, 2]^{3}$$ | |
| 2.5.1.0a1.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ | |
|
\(7\)
| 7.5.2.5a1.2 | $x^{10} + 2 x^{6} + 8 x^{5} + x^{2} + 8 x + 23$ | $2$ | $5$ | $5$ | $C_{10}$ | $$[\ ]_{2}^{5}$$ |
|
\(6329\)
| $\Q_{6329}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{6329}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{6329}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |