Normalized defining polynomial
\( x^{6} - 3x^{5} + 5x^{3} - 3x + 1 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[0, 3]$ |
| |
| Discriminant: |
\(-34992\)
\(\medspace = -\,2^{4}\cdot 3^{7}\)
|
| |
| Root discriminant: | \(5.72\) |
| |
| Galois root discriminant: | $2^{2/3}3^{7/6}\approx 5.71910575798162$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $S_3$ |
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
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: |
\( -2 a^{5} + 5 a^{4} + 2 a^{3} - 8 a^{2} - 4 a + 4 \)
(order $6$)
|
| |
| Fundamental units: |
$a^{5}-2a^{4}-2a^{3}+4a^{2}+2a-2$, $a^{5}-3a^{4}+4a^{2}+a-1$
|
| |
| Regulator: | \( 1.81542571879 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 1.81542571879 \cdot 1}{6\cdot\sqrt{34992}}\cr\approx \mathstrut & 0.401219890968 \end{aligned}\]
Galois group
| A solvable group of order 6 |
| The 3 conjugacy class representatives for $S_3$ |
| Character table for $S_3$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.108.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 3.1.108.1 $\times$ \(\Q\) $\times$ \(\Q\) $\times$ \(\Q\) |
| Degree 3 sibling: | 3.1.108.1 |
| Minimal sibling: | 3.1.108.1 |
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 | R | R | ${\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.4a1.2 | $x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
|
\(3\)
| 3.1.6.7a1.1 | $x^{6} + 3 x^{2} + 3$ | $6$ | $1$ | $7$ | $S_3$ | $$[\frac{3}{2}]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *2 | 2.108.3t2.a.a | $2$ | $ 2^{2} \cdot 3^{3}$ | 6.0.34992.1 | $S_3$ (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 *.