Normalized defining polynomial
\( x^{6} - x^{5} - 10x^{4} + 11x^{3} + 23x^{2} - 44x + 64 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[0, 3]$ |
| |
| Discriminant: |
\(-45696175\)
\(\medspace = -\,5^{2}\cdot 7^{3}\cdot 73^{2}\)
|
| |
| Root discriminant: | \(18.91\) |
| |
| Galois root discriminant: | $5^{1/2}7^{1/2}73^{1/2}\approx 50.547007824400445$ | ||
| Ramified primes: |
\(5\), \(7\), \(73\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-7}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a$, $\frac{1}{164}a^{5}-\frac{13}{164}a^{4}-\frac{9}{82}a^{3}+\frac{63}{164}a^{2}-\frac{77}{164}a+\frac{15}{41}$
| Monogenic: | No | |
| Index: | $4$ | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{319}{164}a^{5}+\frac{142}{41}a^{4}-\frac{431}{41}a^{3}-\frac{1715}{164}a^{2}+\frac{1187}{82}a-\frac{1775}{41}$, $\frac{88}{41}a^{5}+\frac{295}{82}a^{4}-\frac{518}{41}a^{3}-\frac{483}{41}a^{2}+\frac{1413}{82}a-\frac{2059}{41}$
|
| |
| Regulator: | \( 125.162266004 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 125.162266004 \cdot 1}{2\cdot\sqrt{45696175}}\cr\approx \mathstrut & 2.29637654397 \end{aligned}\]
Galois group
| A solvable group of order 12 |
| The 6 conjugacy class representatives for $D_{6}$ |
| Character table for $D_{6}$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), 3.1.2555.1 |
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: | 3.1.2555.1 $\times$ \(\Q(\sqrt{365}) \) $\times$ \(\Q\) |
| Degree 6 sibling: | 6.2.2382729125.1 |
| 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 | ${\href{/padicField/2.2.0.1}{2} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }$ | R | R | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(7\)
| 7.1.2.1a1.1 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 7.1.2.1a1.1 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 7.1.2.1a1.1 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(73\)
| 73.2.1.0a1.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 73.2.2.2a1.2 | $x^{4} + 140 x^{3} + 4910 x^{2} + 700 x + 98$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *12 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.2555.2t1.a.a | $1$ | $ 5 \cdot 7 \cdot 73 $ | \(\Q(\sqrt{-2555}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.365.2t1.a.a | $1$ | $ 5 \cdot 73 $ | \(\Q(\sqrt{365}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *12 | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *12 | 2.2555.3t2.a.a | $2$ | $ 5 \cdot 7 \cdot 73 $ | 3.1.2555.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| *12 | 2.2555.6t3.a.a | $2$ | $ 5 \cdot 7 \cdot 73 $ | 6.0.45696175.1 | $D_{6}$ (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 *.