Normalized defining polynomial
\( x^{6} - 8x^{3} + 25x^{2} - 40x + 32 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[0, 3]$ |
| |
| Discriminant: |
\(-8479744\)
\(\medspace = -\,2^{10}\cdot 7^{2}\cdot 13^{2}\)
|
| |
| Root discriminant: | \(14.28\) |
| |
| Galois root discriminant: | $2^{2}7^{1/2}13^{1/2}\approx 38.157568056677825$ | ||
| Ramified primes: |
\(2\), \(7\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-1}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{436}a^{5}+\frac{22}{109}a^{4}-\frac{26}{109}a^{3}-\frac{1}{109}a^{2}+\frac{1}{4}a-\frac{10}{109}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: |
\( -\frac{25}{436} a^{5} - \frac{5}{109} a^{4} - \frac{4}{109} a^{3} + \frac{25}{109} a^{2} - \frac{5}{4} a + \frac{141}{109} \)
(order $4$)
|
| |
| Fundamental units: |
$\frac{66}{109}a^{5}+\frac{31}{109}a^{4}-\frac{106}{109}a^{3}-\frac{700}{109}a^{2}+13a-\frac{569}{109}$, $\frac{77}{109}a^{5}+\frac{236}{109}a^{4}+\frac{385}{109}a^{3}-\frac{308}{109}a^{2}-\frac{573}{109}$
|
| |
| Regulator: | \( 72.1002516578 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 72.1002516578 \cdot 1}{4\cdot\sqrt{8479744}}\cr\approx \mathstrut & 1.53541232941 \end{aligned}\]
Galois group
| A solvable group of order 12 |
| The 6 conjugacy class representatives for $D_{6}$ |
| Character table for $D_{6}$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 3.1.728.1 |
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: | 3.1.728.1 $\times$ \(\Q(\sqrt{182}) \) $\times$ \(\Q\) |
| Degree 6 sibling: | 6.2.1543313408.3 |
| 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 | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.4.8b1.2 | $x^{4} + 2 x^{2} + 4 x + 10$ | $4$ | $1$ | $8$ | $C_2^2$ | $$[2, 3]$$ | |
|
\(7\)
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 7.2.2.2a1.2 | $x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.728.2t1.b.a | $1$ | $ 2^{3} \cdot 7 \cdot 13 $ | \(\Q(\sqrt{-182}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.728.2t1.a.a | $1$ | $ 2^{3} \cdot 7 \cdot 13 $ | \(\Q(\sqrt{182}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 2.728.3t2.a.a | $2$ | $ 2^{3} \cdot 7 \cdot 13 $ | 3.1.728.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.2912.6t3.e.a | $2$ | $ 2^{5} \cdot 7 \cdot 13 $ | 6.0.8479744.1 | $D_{6}$ (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 *.