Normalized defining polynomial
\( x^{21} + 2x - 2 \)
Invariants
| Degree: | $21$ |
| |
| Signature: | $[1, 10]$ |
| |
| Discriminant: |
\(6346298850946300008339733920874496\)
\(\medspace = 2^{20}\cdot 6052302218385982521381124421\)
|
| |
| Root discriminant: | \(40.70\) |
| |
| Galois root discriminant: | $2^{20/21}6052302218385982521381124421^{1/2}\approx 150541256661719.56$ | ||
| Ramified primes: |
\(2\), \(6052302218385982521381124421\)
|
| |
| Discriminant root field: | $\Q(\sqrt{60523\!\cdots\!24421}$) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$
| 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: | $10$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a-1$, $a^{4}-a^{2}+1$, $a^{14}+a^{7}-a+1$, $a^{16}+a^{11}-a+1$, $a^{14}+a^{11}-a^{5}+a^{3}-a^{2}+a+1$, $a^{20}+a^{17}+2a^{16}+2a^{15}+a^{14}+a^{11}+2a^{10}+a^{9}-a^{8}-2a^{7}-a^{6}+a^{5}+a^{4}-a^{3}-3a^{2}-2a+3$, $a^{20}+a^{13}+a^{6}-a^{5}+a^{3}-a+1$, $2a^{20}+2a^{19}+2a^{18}+2a^{17}+2a^{16}+2a^{15}+a^{14}+a^{13}+a^{12}-a^{9}-a^{8}-a^{7}-a^{6}-2a^{5}-2a^{4}-2a^{2}-a+3$, $a^{20}+2a^{19}+2a^{18}+2a^{17}+2a^{16}+2a^{15}+a^{14}+a^{11}+a^{10}-a^{8}-a^{7}-a^{6}-2a^{5}-2a^{4}-a^{3}+a^{2}+1$, $4a^{20}+a^{19}+3a^{18}+2a^{17}+a^{16}+3a^{15}+3a^{13}+2a^{11}+a^{10}+3a^{8}-2a^{7}+4a^{6}-2a^{5}+2a^{4}+a^{3}-2a^{2}+5a+3$
|
| |
| Regulator: | \( 586206007.678 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 586206007.678 \cdot 1}{2\cdot\sqrt{6346298850946300008339733920874496}}\cr\approx \mathstrut & 0.705648932413 \end{aligned}\] (assuming GRH)
Galois group
| A non-solvable group of order 51090942171709440000 |
| The 792 conjugacy class representatives for $S_{21}$ |
| Character table for $S_{21}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 42 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 | R | ${\href{/padicField/3.11.0.1}{11} }{,}\,{\href{/padicField/3.10.0.1}{10} }$ | $16{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.13.0.1}{13} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.9.0.1}{9} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.21.20a1.1 | $x^{21} + 2$ | $21$ | $1$ | $20$ | 21T11 | $$[\ ]_{21}^{6}$$ |
|
\(605\!\cdots\!421\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $$[\ ]^{7}$$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $$[\ ]^{12}$$ |