Normalized defining polynomial
\( x^{22} - x - 1 \)
Invariants
| Degree: | $22$ |
| |
| Signature: | $[2, 10]$ |
| |
| Discriminant: |
\(347270464382605539918027848005\)
\(\medspace = 5\cdot 69454092876521107983605569601\)
|
| |
| Root discriminant: | \(22.02\) |
| |
| Galois root discriminant: | $5^{1/2}69454092876521107983605569601^{1/2}\approx 589296584397538.8$ | ||
| Ramified primes: |
\(5\), \(69454092876521107983605569601\)
|
| |
| Discriminant root field: | $\Q(\sqrt{34727\!\cdots\!48005}$) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | Trivial group, which has order $1$ (assuming GRH) |
|
Unit group
| Rank: | $11$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a$, $a^{21}-a^{20}-a^{9}-1$, $a^{10}-a^{3}$, $a^{21}-a^{20}+a^{19}-a^{18}+a^{17}-a^{16}-a^{14}-1$, $a^{9}-a^{6}$, $a^{5}+a$, $a^{21}+a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-2$, $a^{21}-a^{20}+a^{17}-a^{16}-1$, $a^{11}-a^{8}+a^{5}$, $a^{21}-a^{20}+a^{19}-a^{18}+a^{17}-a^{16}+a^{15}-a^{14}+a^{13}-a^{3}+a^{2}-a-1$, $a^{21}-a^{20}+a^{19}-a^{18}+a^{17}+a^{15}-a^{14}-a^{12}+a^{9}-1$
|
| |
| Regulator: | \( 912353.207873 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 912353.207873 \cdot 1}{2\cdot\sqrt{347270464382605539918027848005}}\cr\approx \mathstrut & 0.296932516356 \end{aligned}\] (assuming GRH)
Galois group
| A non-solvable group of order 1124000727777607680000 |
| The 1002 conjugacy class representatives for $S_{22}$ |
| Character table for $S_{22}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 44 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 | $22$ | $17{,}\,{\href{/padicField/3.5.0.1}{5} }$ | R | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $16{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.7.0.1}{7} }^{2}{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $21{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $22$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | $21{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.3.1.0a1.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 5.4.1.0a1.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 5.11.1.0a1.1 | $x^{11} + 3 x + 3$ | $1$ | $11$ | $0$ | $C_{11}$ | $$[\ ]^{11}$$ | |
|
\(694\!\cdots\!601\)
| $\Q_{69\!\cdots\!01}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{69\!\cdots\!01}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $$[\ ]^{18}$$ |