Normalized defining polynomial
\( x^{6} - 3x^{5} + 5x^{4} - 5x^{3} + 5x^{2} - 3x + 1 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(0, 3)$ |
| |
| Discriminant: |
\(-12167\)
\(\medspace = -\,23^{3}\)
|
| |
| Root discriminant: | \(4.80\) |
| |
| Galois root discriminant: | $23^{1/2}\approx 4.795831523312719$ | ||
| Ramified primes: |
\(23\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $S_3$ |
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-23}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{5}-2a^{4}+2a^{3}-a^{2}+2a$, $a^{4}-2a^{3}+3a^{2}-2a+2$
|
| |
| Regulator: | \( 0.237219601798 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 0.237219601798 \cdot 1}{2\cdot\sqrt{12167}}\cr\approx \mathstrut & 0.2667278890948 \end{aligned}\]
Galois group
| A solvable group of order 6 |
| The 3 conjugacy class representatives for $S_3$ |
| Character table for $S_3$ |
Intermediate fields
| \(\Q(\sqrt{-23}) \), 3.1.23.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 3.1.23.1 $\times$ \(\Q\) $\times$ \(\Q\) $\times$ \(\Q\) |
| Degree 3 sibling: | 3.1.23.1 |
| Minimal sibling: | 3.1.23.1 |
Multiplicative Galois module structure
| $U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $A$ |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.1.0.1}{1} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.1.2.1a1.1 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 23.1.2.1a1.1 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 23.1.2.1a1.1 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
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) | $1$ | $-1$ |
| *12 | 2.23.3t2.a.a | $2$ | $ 23 $ | 6.0.12167.1 | $S_3$ (as 6T2) | $1$ | $0$ |
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 *.