Normalized defining polynomial
\( x^{6} - 2x^{5} - 53x^{4} + 42x^{3} + 446x^{2} - 836x + 401 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(6, 0)$ |
| |
| Discriminant: |
\(119946304000\)
\(\medspace = 2^{9}\cdot 5^{3}\cdot 37^{4}\)
|
| |
| Root discriminant: | \(70.23\) |
| |
| Galois root discriminant: | $2^{3/2}5^{1/2}37^{2/3}\approx 70.22598052313695$ | ||
| Ramified primes: |
\(2\), \(5\), \(37\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{10}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_6$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1480=2^{3}\cdot 5\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1480}(1,·)$, $\chi_{1480}(149,·)$, $\chi_{1480}(121,·)$, $\chi_{1480}(1321,·)$, $\chi_{1480}(269,·)$, $\chi_{1480}(1469,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{1009}a^{5}-\frac{323}{1009}a^{4}-\frac{297}{1009}a^{3}-\frac{476}{1009}a^{2}-\frac{126}{1009}a+\frac{259}{1009}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{6}$, which has order $6$ |
| |
| Narrow class group: | $C_{6}$, which has order $6$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{50}{1009}a^{5}-\frac{6}{1009}a^{4}-\frac{2742}{1009}a^{3}-\frac{2611}{1009}a^{2}+\frac{19934}{1009}a-\frac{14293}{1009}$, $\frac{166}{1009}a^{5}-\frac{141}{1009}a^{4}-\frac{8942}{1009}a^{3}-\frac{3341}{1009}a^{2}+\frac{69894}{1009}a-\frac{56897}{1009}$, $\frac{166}{1009}a^{5}-\frac{141}{1009}a^{4}-\frac{8942}{1009}a^{3}-\frac{3341}{1009}a^{2}+\frac{68885}{1009}a-\frac{55888}{1009}$, $\frac{116}{1009}a^{5}-\frac{135}{1009}a^{4}-\frac{6200}{1009}a^{3}-\frac{730}{1009}a^{2}+\frac{48951}{1009}a-\frac{46640}{1009}$, $\frac{50}{1009}a^{5}-\frac{6}{1009}a^{4}-\frac{2742}{1009}a^{3}-\frac{2611}{1009}a^{2}+\frac{18925}{1009}a-\frac{13284}{1009}$
|
| |
| Regulator: | \( 357.158964391 \) |
| |
| Unit signature rank: | \( 6 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 357.158964391 \cdot 6}{2\cdot\sqrt{119946304000}}\cr\approx \mathstrut & 0.198001895846 \end{aligned}\]
Galois group
| A cyclic group of order 6 |
| The 6 conjugacy class representatives for $C_6$ |
| Character table for $C_6$ |
Intermediate fields
| \(\Q(\sqrt{10}) \), 3.3.1369.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{10}) \) $\times$ 3.3.1369.1 |
| 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 | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.1.0.1}{1} }^{6}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.9a1.6 | $x^{6} + 2 x^{4} + 6 x^{3} + x^{2} + 6 x + 15$ | $2$ | $3$ | $9$ | $C_6$ | $$[3]^{3}$$ |
|
\(5\)
| 5.3.2.3a1.1 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 23 x + 9$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
|
\(37\)
| 37.1.3.2a1.1 | $x^{3} + 37$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 37.1.3.2a1.1 | $x^{3} + 37$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
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.a.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{10}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *6 | 1.37.3t1.a.a | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.1480.6t1.b.a | $1$ | $ 2^{3} \cdot 5 \cdot 37 $ | 6.6.119946304000.2 | $C_6$ (as 6T1) | $0$ | $1$ |
| *6 | 1.37.3t1.a.b | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.1480.6t1.b.b | $1$ | $ 2^{3} \cdot 5 \cdot 37 $ | 6.6.119946304000.2 | $C_6$ (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 *.