Normalized defining polynomial
\( x^{6} - 3x^{5} + 12x^{4} - 17x^{3} + 87x^{2} - 114x + 323 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(0, 3)$ |
| |
| Discriminant: |
\(-45001899\)
\(\medspace = -\,3^{8}\cdot 19^{3}\)
|
| |
| Root discriminant: | \(18.86\) |
| |
| Galois root discriminant: | $3^{4/3}19^{1/2}\approx 18.859860385004858$ | ||
| Ramified primes: |
\(3\), \(19\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_6$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(171=3^{2}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{171}(1,·)$, $\chi_{171}(115,·)$, $\chi_{171}(37,·)$, $\chi_{171}(151,·)$, $\chi_{171}(58,·)$, $\chi_{171}(94,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-19}) \), 6.0.45001899.1$^{3}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{14977}a^{5}-\frac{5789}{14977}a^{4}+\frac{6594}{14977}a^{3}-\frac{6482}{14977}a^{2}+\frac{2531}{14977}a+\frac{178}{881}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}$, which has order $4$ |
| |
| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ |
| |
| Relative class number: | $4$ |
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{150}{14977}a^{5}+\frac{316}{14977}a^{4}+\frac{618}{14977}a^{3}+\frac{1205}{14977}a^{2}+\frac{5225}{14977}a+\frac{1151}{881}$, $\frac{132}{14977}a^{5}-\frac{321}{14977}a^{4}+\frac{1742}{14977}a^{3}-\frac{1935}{14977}a^{2}+\frac{4598}{14977}a-\frac{291}{881}$
|
| |
| Regulator: | \( 3.39714980258 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 3.39714980258 \cdot 4}{2\cdot\sqrt{45001899}}\cr\approx \mathstrut & 0.251228484543 \end{aligned}\]
Galois group
| A cyclic group of order 6 |
| The 6 conjugacy class representatives for $C_6$ |
| Character table for $C_6$ |
Intermediate fields
| \(\Q(\sqrt{-19}) \), \(\Q(\zeta_{9})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | \(\Q(\zeta_{9})^+\) $\times$ \(\Q(\sqrt{-19}) \) $\times$ \(\Q\) |
| 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 | ${\href{/padicField/2.6.0.1}{6} }$ | R | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.1.0.1}{1} }^{6}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.3.8a2.1 | $x^{6} + 6 x^{5} + 24 x^{4} + 56 x^{3} + 84 x^{2} + 72 x + 35$ | $3$ | $2$ | $8$ | $C_6$ | $$[2]^{2}$$ |
|
\(19\)
| 19.1.2.1a1.1 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 19.1.2.1a1.1 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 19.1.2.1a1.1 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
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 | 1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *6 | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.171.6t1.f.a | $1$ | $ 3^{2} \cdot 19 $ | 6.0.45001899.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| *6 | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.171.6t1.f.b | $1$ | $ 3^{2} \cdot 19 $ | 6.0.45001899.1 | $C_6$ (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 *.