Normalized defining polynomial
\( x^{6} - 2x^{5} + 19x^{4} - 14x^{3} + 342x^{2} - 584x + 2849 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(0, 3)$ |
| |
| Discriminant: |
\(-8340544000\)
\(\medspace = -\,2^{9}\cdot 5^{3}\cdot 19^{4}\)
|
| |
| Root discriminant: | \(45.03\) |
| |
| Galois root discriminant: | $2^{3/2}5^{1/2}19^{2/3}\approx 45.0331572624958$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-10}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_6$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(760=2^{3}\cdot 5\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{760}(539,·)$, $\chi_{760}(1,·)$, $\chi_{760}(419,·)$, $\chi_{760}(121,·)$, $\chi_{760}(201,·)$, $\chi_{760}(619,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-10}) \), 6.0.8340544000.3$^{3}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{139601}a^{5}-\frac{35655}{139601}a^{4}+\frac{1028}{139601}a^{3}+\frac{63765}{139601}a^{2}-\frac{10918}{139601}a+\frac{18}{49}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{78}$, which has order $78$ |
| |
| Narrow class group: | $C_{78}$, which has order $78$ |
| |
| Relative class number: | $78$ |
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{274}{139601}a^{5}+\frac{2600}{139601}a^{4}+\frac{2470}{139601}a^{3}+\frac{21485}{139601}a^{2}+\frac{79690}{139601}a+\frac{179}{49}$, $\frac{552}{139601}a^{5}+\frac{2181}{139601}a^{4}+\frac{9052}{139601}a^{3}+\frac{18828}{139601}a^{2}+\frac{115708}{139601}a+\frac{87}{49}$
|
| |
| Regulator: | \( 7.80862678603 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 7.80862678603 \cdot 78}{2\cdot\sqrt{8340544000}}\cr\approx \mathstrut & 0.827145077250 \end{aligned}\]
Galois group
| A cyclic group of order 6 |
| The 6 conjugacy class representatives for $C_6$ |
| Character table for $C_6$ |
Intermediate fields
| \(\Q(\sqrt{-10}) \), 3.3.361.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{-10}) \) $\times$ 3.3.361.1 |
| 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 | R | ${\href{/padicField/3.6.0.1}{6} }$ | R | ${\href{/padicField/7.1.0.1}{1} }^{6}$ | ${\href{/padicField/11.1.0.1}{1} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }$ | R | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.9a1.2 | $x^{6} + 2 x^{4} + 2 x^{3} + x^{2} + 2 x + 11$ | $2$ | $3$ | $9$ | $C_6$ | $$[3]^{3}$$ |
|
\(5\)
| 5.3.2.3a1.1 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 23 x + 9$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
|
\(19\)
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
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 | 1.40.2t1.b.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{-10}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *6 | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.760.6t1.b.a | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.8340544000.3 | $C_6$ (as 6T1) | $0$ | $-1$ |
| *6 | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.760.6t1.b.b | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.8340544000.3 | $C_6$ (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 *.