Normalized defining polynomial
\( x^{6} - 2x^{5} - x^{4} - 2x^{3} + 34x^{2} + 28x + 73 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(0, 3)$ |
| |
| Discriminant: |
\(-14623232\)
\(\medspace = -\,2^{9}\cdot 13^{4}\)
|
| |
| Root discriminant: | \(15.64\) |
| |
| Galois root discriminant: | $2^{3/2}13^{2/3}\approx 15.637736649622886$ | ||
| Ramified primes: |
\(2\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_6$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(104=2^{3}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{104}(1,·)$, $\chi_{104}(3,·)$, $\chi_{104}(35,·)$, $\chi_{104}(81,·)$, $\chi_{104}(9,·)$, $\chi_{104}(27,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-2}) \), 6.0.14623232.1$^{3}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3697}a^{5}+\frac{48}{3697}a^{4}-\frac{1298}{3697}a^{3}+\frac{1644}{3697}a^{2}+\frac{900}{3697}a+\frac{664}{3697}$
| Monogenic: | Not computed | |
| 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: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{74}{3697}a^{5}-\frac{145}{3697}a^{4}+\frac{70}{3697}a^{3}-\frac{345}{3697}a^{2}+\frac{54}{3697}a+\frac{1075}{3697}$, $\frac{74}{3697}a^{5}-\frac{145}{3697}a^{4}+\frac{70}{3697}a^{3}-\frac{345}{3697}a^{2}+\frac{54}{3697}a-\frac{2622}{3697}$
|
| |
| Regulator: | \( 5.46019947038 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 5.46019947038 \cdot 1}{2\cdot\sqrt{14623232}}\cr\approx \mathstrut & 0.177090976550 \end{aligned}\]
Galois group
| A cyclic group of order 6 |
| The 6 conjugacy class representatives for $C_6$ |
| Character table for $C_6$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 3.3.169.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{-2}) \) $\times$ 3.3.169.1 |
| 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 | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.9a1.1 | $x^{6} + 2 x^{4} + 2 x^{3} + x^{2} + 2 x + 3$ | $2$ | $3$ | $9$ | $C_6$ | $$[3]^{3}$$ |
|
\(13\)
| 13.2.3.4a1.2 | $x^{6} + 36 x^{5} + 438 x^{4} + 1872 x^{3} + 876 x^{2} + 144 x + 21$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ |
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 | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *6 | 1.13.3t1.a.a | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.104.6t1.d.a | $1$ | $ 2^{3} \cdot 13 $ | 6.0.14623232.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| *6 | 1.13.3t1.a.b | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.104.6t1.d.b | $1$ | $ 2^{3} \cdot 13 $ | 6.0.14623232.1 | $C_6$ (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 *.