Normalized defining polynomial
\( x^{6} - x^{5} - 6x^{4} + 6x^{3} + 8x^{2} - 8x + 1 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(6, 0)$ |
| |
| Discriminant: |
\(453789\)
\(\medspace = 3^{3}\cdot 7^{5}\)
|
| |
| Root discriminant: | \(8.77\) |
| |
| Galois root discriminant: | $3^{1/2}7^{5/6}\approx 8.766151944295878$ | ||
| Ramified primes: |
\(3\), \(7\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{21}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_6$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(21=3\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{21}(16,·)$, $\chi_{21}(1,·)$, $\chi_{21}(4,·)$, $\chi_{21}(5,·)$, $\chi_{21}(17,·)$, $\chi_{21}(20,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{5}-5a^{3}+a^{2}+5a-2$, $a^{3}-3a$, $a^{5}-5a^{3}+5a-1$, $a^{3}-4a+1$, $a-1$
|
| |
| Regulator: | \( 4.39996237517 \) |
| |
| Unit signature rank: | \( 5 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 4.39996237517 \cdot 1}{2\cdot\sqrt{453789}}\cr\approx \mathstrut & 0.209012355184 \end{aligned}\]
Galois group
| A cyclic group of order 6 |
| The 6 conjugacy class representatives for $C_6$ |
| Character table for $C_6$ |
Intermediate fields
| \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{21}) \) $\times$ \(\Q(\zeta_{7})^+\) |
| 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 | ${\href{/padicField/2.6.0.1}{6} }$ | R | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.2.3a1.1 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
|
\(7\)
| 7.1.6.5a1.1 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |
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 | 1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *6 | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.21.6t1.b.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\zeta_{21})^+\) | $C_6$ (as 6T1) | $0$ | $1$ |
| *6 | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.21.6t1.b.b | $1$ | $ 3 \cdot 7 $ | \(\Q(\zeta_{21})^+\) | $C_6$ (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 *.