Normalized defining polynomial
\( x^{22} - x^{11} + 3 \)
Invariants
| Degree: | $22$ |
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| Signature: | $[0, 11]$ |
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| Discriminant: |
\(-1371422143339955182918013386604257876419\)
\(\medspace = -\,3^{10}\cdot 11^{33}\)
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| Root discriminant: | \(60.11\) |
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| Galois root discriminant: | $3^{10/11}11^{169/110}\approx 108.06982491208107$ | ||
| Ramified primes: |
\(3\), \(11\)
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| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-11}) \) | ||
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) |
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| Narrow class group: | Trivial group, which has order $1$ (assuming GRH) |
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Unit group
| Rank: | $10$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$a^{20}+2a^{19}+2a^{18}+a^{17}+a^{14}+2a^{13}+2a^{12}+a^{11}-a^{9}-a^{8}+a^{6}+a^{5}-a^{3}-a^{2}+1$, $a^{21}-a^{18}+2a^{16}+a^{15}-a^{13}-3a^{12}-a^{11}+a^{9}+a^{8}-2a^{6}-2a^{5}+2a^{4}+3a^{3}+3a^{2}+2a-4$, $6a^{21}+2a^{20}-8a^{19}+2a^{18}+8a^{17}-6a^{16}-5a^{15}+10a^{14}+a^{13}-11a^{12}+5a^{11}+3a^{10}-13a^{9}+3a^{8}+12a^{7}-11a^{6}-9a^{5}+16a^{4}+a^{3}-18a^{2}+10a+16$, $2a^{19}-2a^{18}-a^{16}+a^{15}+2a^{14}-3a^{13}-3a^{11}+2a^{10}-4a^{8}-2a^{6}+4a^{5}-a^{4}-a^{3}-2a^{2}+2a+4$, $a^{21}-a^{19}+a^{18}-a^{15}+2a^{13}-2a^{12}-a^{11}+2a^{10}-a^{9}-a^{7}+a^{6}+a^{5}-2a^{4}+2a^{3}-a+2$, $a^{21}+4a^{20}-a^{19}-4a^{18}+a^{17}+5a^{16}-a^{15}-5a^{14}+2a^{13}+5a^{12}-2a^{11}-6a^{10}-a^{9}+7a^{8}-7a^{6}+a^{5}+8a^{4}-a^{3}-9a^{2}+2a+11$, $2a^{20}+2a^{19}-a^{18}-3a^{17}-a^{16}+3a^{15}+3a^{14}-2a^{13}-4a^{12}-a^{11}+5a^{10}+a^{9}-5a^{8}-5a^{7}+4a^{6}+6a^{5}+a^{4}-7a^{3}-4a^{2}+3a+8$, $a^{21}-4a^{19}+4a^{18}-2a^{17}+4a^{16}-5a^{15}+2a^{13}+a^{12}+4a^{11}-10a^{10}+3a^{9}+a^{8}+5a^{7}+a^{6}-10a^{5}+4a^{4}-4a^{3}+13a^{2}-7a-2$, $a^{20}+3a^{19}+2a^{18}-a^{17}-a^{16}-a^{15}-a^{14}-3a^{13}+3a^{11}+2a^{10}-a^{9}-4a^{8}-a^{7}-2a^{6}-3a^{5}+5a^{3}+7a^{2}+a+1$, $a^{20}-a^{19}-a^{18}+a^{17}-a^{15}+a^{14}+3a^{13}-a^{11}-a^{9}+a^{7}-a^{6}-2a^{5}-a^{3}-3a^{2}+2a+5$
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| Regulator: | \( 107658885493 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 107658885493 \cdot 1}{2\cdot\sqrt{1371422143339955182918013386604257876419}}\cr\approx \mathstrut & 0.875815964202034 \end{aligned}\] (assuming GRH)
Galois group
$C_{11}:F_{11}$ (as 22T11):
| A solvable group of order 1210 |
| The 25 conjugacy class representatives for $C_{11}:F_{11}$ |
| Character table for $C_{11}:F_{11}$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 22 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 | ${\href{/padicField/2.10.0.1}{10} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.5.0.1}{5} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $22$ | ${\href{/padicField/47.5.0.1}{5} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 3.5.1.0a1.1 | $x^{5} + 2 x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ | |
| 3.5.1.0a1.1 | $x^{5} + 2 x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ | |
| 3.1.11.10a1.1 | $x^{11} + 3$ | $11$ | $1$ | $10$ | $C_{11}:C_5$ | $$[\ ]_{11}^{5}$$ | |
|
\(11\)
| 11.1.22.33a1.1 | $x^{22} + 22 x^{12} + 11$ | $22$ | $1$ | $33$ | 22T5 | not computed |