Normalized defining polynomial
\( x^{23} - 3 \)
Invariants
| Degree: | $23$ |
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| Signature: | $[1, 11]$ |
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| Discriminant: |
\(-655251210967044330484283743635659792988303\)
\(\medspace = -\,3^{22}\cdot 23^{23}\)
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| Root discriminant: | \(65.78\) |
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| Galois root discriminant: | $3^{22/23}23^{527/506}\approx 74.92367730844663$ | ||
| Ramified primes: |
\(3\), \(23\)
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| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$
| 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: | $11$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$a^{12}-a-1$, $a^{22}+a^{21}+a^{20}+a^{19}+a^{17}-a^{14}-a^{13}-a^{12}-2a^{11}-a^{10}-2a^{9}-a^{8}-2a^{7}-a^{6}+a^{3}+3a+2$, $a^{22}+a^{21}+a^{20}+a^{19}+a^{17}+a^{16}-a^{15}-a^{12}-a^{10}-a^{9}-a^{7}+a^{5}+a^{3}+3a^{2}+2a+1$, $a^{22}+a^{15}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}-a^{7}-a^{5}-a^{4}+a^{2}+2a+1$, $a^{18}+a^{16}-a^{15}+2a^{14}-a^{13}+2a^{12}-2a^{11}+2a^{10}-2a^{9}+2a^{8}-3a^{7}+2a^{6}-3a^{5}+2a^{4}-3a^{3}+a^{2}-2a+1$, $a^{22}+a^{21}-2a^{20}+2a^{16}+a^{15}-a^{14}-a^{13}-2a^{12}-4a^{11}+a^{10}+3a^{7}-2a^{6}-2a^{5}-a^{4}-2a^{3}+a^{2}+6a+2$, $a^{22}+a^{20}-a^{16}+a^{15}-a^{14}+3a^{13}-a^{12}+3a^{11}-3a^{10}+2a^{9}-3a^{8}+3a^{7}-a^{6}+4a^{5}-a^{4}+2a^{3}-3a^{2}+a-2$, $a^{22}+a^{21}-2a^{19}-2a^{18}-2a^{17}+a^{15}+4a^{14}+3a^{13}+2a^{12}-2a^{11}-2a^{10}-4a^{9}-2a^{8}-a^{7}+3a^{6}+2a^{5}+a^{4}-a^{3}+1$, $2a^{21}-2a^{20}+a^{19}+a^{18}-3a^{17}+2a^{16}-a^{15}-2a^{14}+a^{13}-3a^{11}+3a^{10}-a^{9}-2a^{8}+4a^{7}-2a^{6}-a^{5}+5a^{4}-4a^{3}+2a^{2}+3a-2$, $a^{22}-2a^{20}+a^{19}+2a^{17}-2a^{16}-a^{15}+a^{14}+a^{13}+a^{12}-4a^{11}+a^{10}+a^{9}+3a^{8}-2a^{7}-3a^{6}+2a^{5}+a^{4}+3a^{3}-5a^{2}+a+1$, $4a^{22}-a^{21}-5a^{20}-7a^{19}-5a^{18}+3a^{17}+9a^{16}+10a^{15}+5a^{14}-4a^{13}-10a^{12}-9a^{11}-5a^{10}+4a^{9}+13a^{8}+13a^{7}+4a^{6}-11a^{5}-22a^{4}-16a^{3}+a^{2}+14a+20$
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| Regulator: | \( 1289649122770 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 1289649122770 \cdot 1}{2\cdot\sqrt{655251210967044330484283743635659792988303}}\cr\approx \mathstrut & 0.959944672896699 \end{aligned}\] (assuming GRH)
Galois group
| A solvable group of order 506 |
| The 23 conjugacy class representatives for $F_{23}$ |
| Character table for $F_{23}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 46 sibling: | deg 46 |
| 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 | ${\href{/padicField/2.11.0.1}{11} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | $22{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/19.1.0.1}{1} }$ | R | ${\href{/padicField/29.11.0.1}{11} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.11.0.1}{11} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.11.0.1}{11} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $23$ | $22{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.11.0.1}{11} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.23.22a1.1 | $x^{23} + 3$ | $23$ | $1$ | $22$ | $C_{23}:C_{11}$ | $$[\ ]_{23}^{11}$$ |
|
\(23\)
| 23.1.23.23a1.1 | $x^{23} + 23 x + 23$ | $23$ | $1$ | $23$ | $F_{23}$ | $$[\frac{23}{22}]_{22}$$ |