Normalized defining polynomial
\( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(0, 3)$ |
| |
| Discriminant: |
\(-16807\)
\(\medspace = -\,7^{5}\)
|
| |
| Root discriminant: | \(5.06\) |
| |
| Galois root discriminant: | $7^{5/6}\approx 5.061140184796386$ | ||
| Ramified primes: |
\(7\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_6$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{7}(1,·)$, $\chi_{7}(2,·)$, $\chi_{7}(3,·)$, $\chi_{7}(4,·)$, $\chi_{7}(5,·)$, $\chi_{7}(6,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})\)$^{3}$ | ||
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 $1$ |
| |
| Relative class number: | $1$ |
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( a \)
(order $14$)
|
| |
| Fundamental units: |
$a-1$, $a^{4}-a$
|
| |
| Regulator: | \( 2.10181872849 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 2.10181872849 \cdot 1}{14\cdot\sqrt{16807}}\cr\approx \mathstrut & 0.287251117499 \end{aligned}\]
Galois group
| A cyclic group of order 6 |
| The 6 conjugacy class representatives for $C_6$ |
| Character table for $C_6$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | \(\Q(\zeta_{7})^+\) $\times$ \(\Q(\sqrt{-7}) \) $\times$ \(\Q\) |
| 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 | ${\href{/padicField/2.3.0.1}{3} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.6.0.1}{6} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 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 | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *6 | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.7.6t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})\) | $C_6$ (as 6T1) | $0$ | $-1$ |
| *6 | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.7.6t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})\) | $C_6$ (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 *.