Normalized defining polynomial
\( x^{6} - 2x^{5} - 3x^{4} - 8x^{3} + 135x^{2} - 214x + 155 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(0, 3)$ |
| |
| Discriminant: |
\(-1321164656\)
\(\medspace = -\,2^{4}\cdot 7^{5}\cdot 17^{3}\)
|
| |
| Root discriminant: | \(33.13\) |
| |
| Galois root discriminant: | $2\cdot 7^{5/6}17^{1/2}\approx 41.735231135947174$ | ||
| Ramified primes: |
\(2\), \(7\), \(17\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-119}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-119}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{16}a^{4}-\frac{1}{2}a+\frac{7}{16}$, $\frac{1}{64}a^{5}+\frac{1}{64}a^{4}-\frac{1}{8}a^{2}-\frac{17}{64}a-\frac{9}{64}$
| Monogenic: | No | |
| Index: | $4$ | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{30}$, which has order $30$ |
| |
| Narrow class group: | $C_{30}$, which has order $30$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{64}a^{5}-\frac{3}{64}a^{4}-\frac{1}{8}a^{2}+\frac{79}{64}a-\frac{613}{64}$, $\frac{5}{64}a^{5}+\frac{1}{64}a^{4}-\frac{1}{2}a^{3}-\frac{9}{8}a^{2}+\frac{555}{64}a+\frac{151}{64}$
|
| |
| Regulator: | \( 59.5041576386 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 59.5041576386 \cdot 30}{2\cdot\sqrt{1321164656}}\cr\approx \mathstrut & 6.09115571854 \end{aligned}\]
Galois group
| A solvable group of order 12 |
| The 6 conjugacy class representatives for $D_{6}$ |
| Character table for $D_{6}$ |
Intermediate fields
| \(\Q(\sqrt{-119}) \), 3.1.3332.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | deg 12 |
| Twin sextic algebra: | 3.1.3332.1 $\times$ \(\Q(\sqrt{7}) \) $\times$ \(\Q\) |
| Degree 6 sibling: | 6.2.310862272.1 |
| Minimal sibling: | 6.2.310862272.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }$ | R | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
|
\(7\)
| 7.1.6.5a1.6 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |
|
\(17\)
| 17.1.2.1a1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 17.2.2.2a1.2 | $x^{4} + 32 x^{3} + 262 x^{2} + 96 x + 26$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *12 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.68.2t1.a.a | $1$ | $ 2^{2} \cdot 17 $ | \(\Q(\sqrt{-17}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.28.2t1.a.a | $1$ | $ 2^{2} \cdot 7 $ | \(\Q(\sqrt{7}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *12 | 1.119.2t1.a.a | $1$ | $ 7 \cdot 17 $ | \(\Q(\sqrt{-119}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *12 | 2.3332.3t2.a.a | $2$ | $ 2^{2} \cdot 7^{2} \cdot 17 $ | 3.1.3332.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| *12 | 2.3332.6t3.d.a | $2$ | $ 2^{2} \cdot 7^{2} \cdot 17 $ | 6.0.1321164656.1 | $D_{6}$ (as 6T3) | $1$ | $0$ |
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 *.