Normalized defining polynomial
\( x^{6} - 3x^{5} + 15x^{4} - 23x^{3} + 123x^{2} - 153x + 489 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[0, 3]$ |
| |
| Discriminant: |
\(-79827687\)
\(\medspace = -\,3^{8}\cdot 23^{3}\)
|
| |
| Root discriminant: | \(20.75\) |
| |
| Galois root discriminant: | $3^{4/3}23^{1/2}\approx 20.75035786129348$ | ||
| Ramified primes: |
\(3\), \(23\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_6$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(207=3^{2}\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{207}(160,·)$, $\chi_{207}(1,·)$, $\chi_{207}(91,·)$, $\chi_{207}(22,·)$, $\chi_{207}(70,·)$, $\chi_{207}(139,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-23}) \), 6.0.79827687.1$^{3}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{23633}a^{5}+\frac{1361}{23633}a^{4}-\frac{10588}{23633}a^{3}-\frac{2292}{23633}a^{2}-\frac{6609}{23633}a-\frac{10656}{23633}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{9}$, which has order $9$ |
| |
| Narrow class group: | $C_{9}$, which has order $9$ |
| |
| Relative class number: | $9$ |
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{174}{23633}a^{5}+\frac{484}{23633}a^{4}+\frac{1062}{23633}a^{3}+\frac{2953}{23633}a^{2}+\frac{8051}{23633}a+\frac{36496}{23633}$, $\frac{156}{23633}a^{5}-\frac{381}{23633}a^{4}+\frac{2582}{23633}a^{3}-\frac{3057}{23633}a^{2}+\frac{8848}{23633}a-\frac{8026}{23633}$
|
| |
| Regulator: | \( 3.39714980258 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 3.39714980258 \cdot 9}{2\cdot\sqrt{79827687}}\cr\approx \mathstrut & 0.424414334937 \end{aligned}\]
Galois group
| A cyclic group of order 6 |
| The 6 conjugacy class representatives for $C_6$ |
| Character table for $C_6$ |
Intermediate fields
| \(\Q(\sqrt{-23}) \), \(\Q(\zeta_{9})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | \(\Q(\zeta_{9})^+\) $\times$ \(\Q(\sqrt{-23}) \) $\times$ \(\Q\) |
| 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 | ${\href{/padicField/2.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.3.4a2.1 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $$[2]$$ |
| 3.1.3.4a2.1 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $$[2]$$ | |
|
\(23\)
| 23.3.2.3a1.2 | $x^{6} + 4 x^{4} + 36 x^{3} + 4 x^{2} + 72 x + 347$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *6 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *6 | 1.23.2t1.a.a | $1$ | $ 23 $ | \(\Q(\sqrt{-23}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *6 | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.207.6t1.b.a | $1$ | $ 3^{2} \cdot 23 $ | 6.0.79827687.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| *6 | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.207.6t1.b.b | $1$ | $ 3^{2} \cdot 23 $ | 6.0.79827687.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.