Normalized defining polynomial
\( x^{6} + 21x^{4} + 116x^{2} + 64 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[0, 3]$ |
| |
| Discriminant: |
\(-59105344\)
\(\medspace = -\,2^{6}\cdot 31^{4}\)
|
| |
| Root discriminant: | \(19.74\) |
| |
| Galois root discriminant: | $2\cdot 31^{2/3}\approx 19.736544806437948$ | ||
| Ramified primes: |
\(2\), \(31\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_6$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(124=2^{2}\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{124}(1,·)$, $\chi_{124}(67,·)$, $\chi_{124}(5,·)$, $\chi_{124}(87,·)$, $\chi_{124}(25,·)$, $\chi_{124}(63,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-1}) \), 6.0.59105344.1$^{3}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{5}-\frac{3}{16}a^{3}-\frac{1}{4}a$
| Monogenic: | No | |
| Index: | $8$ | |
| Inessential primes: | $2$ |
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: |
\( -\frac{1}{16} a^{5} - \frac{13}{16} a^{3} - \frac{7}{4} a \)
(order $4$)
|
| |
| Fundamental units: |
$\frac{3}{16}a^{5}+\frac{71}{16}a^{3}+\frac{101}{4}a$, $\frac{3}{2}a^{4}+\frac{31}{2}a^{2}+9$
|
| |
| Regulator: | \( 48.7831276503 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 48.7831276503 \cdot 4}{4\cdot\sqrt{59105344}}\cr\approx \mathstrut & 1.57396790141 \end{aligned}\]
Galois group
| A cyclic group of order 6 |
| The 6 conjugacy class representatives for $C_6$ |
| Character table for $C_6$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 3.3.961.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 3.3.961.1 $\times$ \(\Q(\sqrt{-1}) \) $\times$ \(\Q\) |
| 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 | R | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.1.0.1}{1} }^{6}$ | R | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
|
\(31\)
| 31.2.3.4a1.2 | $x^{6} + 87 x^{5} + 2532 x^{4} + 24911 x^{3} + 7596 x^{2} + 783 x + 58$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ |
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 | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *6 | 1.31.3t1.a.a | $1$ | $ 31 $ | 3.3.961.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.124.6t1.a.a | $1$ | $ 2^{2} \cdot 31 $ | 6.0.59105344.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| *6 | 1.31.3t1.a.b | $1$ | $ 31 $ | 3.3.961.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.124.6t1.a.b | $1$ | $ 2^{2} \cdot 31 $ | 6.0.59105344.1 | $C_6$ (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 *.