Normalized defining polynomial
\( x^{10} - x^{9} + 5x^{5} - x^{4} - x^{2} - 3x + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(4, 3)$ |
| |
| Discriminant: |
\(-3531636739\)
\(\medspace = -\,7933\cdot 445183\)
|
| |
| Root discriminant: | \(9.01\) |
| |
| Galois root discriminant: | $7933^{1/2}445183^{1/2}\approx 59427.57557733615$ | ||
| Ramified primes: |
\(7933\), \(445183\)
|
| |
| Discriminant root field: | $\Q(\sqrt{-3531636739}$) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{131}a^{9}+\frac{59}{131}a^{8}+\frac{3}{131}a^{7}+\frac{49}{131}a^{6}+\frac{58}{131}a^{5}-\frac{52}{131}a^{4}+\frac{23}{131}a^{3}-\frac{61}{131}a^{2}+\frac{7}{131}a+\frac{24}{131}$
| 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: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{81}{131}a^{9}-\frac{68}{131}a^{8}-\frac{19}{131}a^{7}+\frac{39}{131}a^{6}-\frac{18}{131}a^{5}+\frac{373}{131}a^{4}+\frac{29}{131}a^{3}-\frac{94}{131}a^{2}+\frac{43}{131}a-\frac{152}{131}$, $\frac{37}{131}a^{9}-\frac{44}{131}a^{8}-\frac{20}{131}a^{7}-\frac{21}{131}a^{6}+\frac{50}{131}a^{5}+\frac{172}{131}a^{4}-\frac{66}{131}a^{3}-\frac{161}{131}a^{2}-\frac{134}{131}a-\frac{29}{131}$, $\frac{37}{131}a^{9}-\frac{44}{131}a^{8}-\frac{20}{131}a^{7}+\frac{110}{131}a^{6}-\frac{81}{131}a^{5}+\frac{172}{131}a^{4}-\frac{66}{131}a^{3}-\frac{30}{131}a^{2}+\frac{259}{131}a-\frac{160}{131}$, $\frac{8}{131}a^{9}-\frac{52}{131}a^{8}+\frac{24}{131}a^{7}-\frac{1}{131}a^{6}+\frac{71}{131}a^{5}-\frac{23}{131}a^{4}-\frac{209}{131}a^{3}-\frac{95}{131}a^{2}-\frac{75}{131}a+\frac{192}{131}$, $a$, $\frac{10}{131}a^{9}-\frac{65}{131}a^{8}+\frac{30}{131}a^{7}-\frac{34}{131}a^{6}+\frac{56}{131}a^{5}+\frac{4}{131}a^{4}-\frac{294}{131}a^{3}-\frac{86}{131}a^{2}-\frac{192}{131}a-\frac{22}{131}$
|
| |
| Regulator: | \( 5.215653226165113 \) |
| |
| Unit signature rank: | \( 4 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 5.215653226165113 \cdot 1}{2\cdot\sqrt{3531636739}}\cr\approx \mathstrut & 0.174160750583968 \end{aligned}\]
Galois group
| A non-solvable group of order 3628800 |
| The 42 conjugacy class representatives for $S_{10}$ |
| Character table for $S_{10}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 20 sibling: | data not computed |
| Degree 45 sibling: | 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 | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.10.0.1}{10} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7933\)
| $\Q_{7933}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{7933}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{7933}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | ||
|
\(445183\)
| $\Q_{445183}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $$[\ ]^{7}$$ |