Normalized defining polynomial
\( x^{6} - 3x^{5} - 2x^{4} + 9x^{3} - 5x^{2} + 21 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[0, 3]$ |
| |
| Discriminant: |
\(-5938947\)
\(\medspace = -\,3^{3}\cdot 7^{2}\cdot 67^{2}\)
|
| |
| Root discriminant: | \(13.46\) |
| |
| Galois root discriminant: | $3^{1/2}7^{1/2}67^{1/2}\approx 37.5099986670221$ | ||
| Ramified primes: |
\(3\), \(7\), \(67\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{21}a^{5}+\frac{8}{21}a^{4}+\frac{2}{21}a^{3}+\frac{10}{21}a^{2}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ |
| |
| Narrow class group: | $C_{3}$, which has order $3$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -\frac{2}{21} a^{5} + \frac{5}{21} a^{4} - \frac{4}{21} a^{3} + \frac{1}{21} a^{2} + a \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{4}{7}a^{5}-\frac{3}{7}a^{4}-\frac{13}{7}a^{3}+\frac{5}{7}a^{2}-2a-5$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{7}{3}a^{3}+\frac{1}{3}a^{2}+5a+3$
|
| |
| Regulator: | \( 21.7255887918 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 21.7255887918 \cdot 3}{6\cdot\sqrt{5938947}}\cr\approx \mathstrut & 1.10567232631 \end{aligned}\]
Galois group
| A solvable group of order 12 |
| The 6 conjugacy class representatives for $D_{6}$ |
| Character table for $D_{6}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.1407.1 |
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: | \(\Q\) $\times$ \(\Q(\sqrt{469}) \) $\times$ 3.1.1407.1 |
| Degree 6 sibling: | 6.2.928455381.1 |
| 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 | ${\href{/padicField/2.6.0.1}{6} }$ | R | ${\href{/padicField/5.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 7.1.2.1a1.2 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 7.1.2.1a1.2 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(67\)
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 67.1.2.1a1.1 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 67.1.2.1a1.1 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *12 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.1407.2t1.a.a | $1$ | $ 3 \cdot 7 \cdot 67 $ | \(\Q(\sqrt{-1407}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.469.2t1.a.a | $1$ | $ 7 \cdot 67 $ | \(\Q(\sqrt{469}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *12 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *12 | 2.1407.3t2.a.a | $2$ | $ 3 \cdot 7 \cdot 67 $ | 3.1.1407.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| *12 | 2.1407.6t3.c.a | $2$ | $ 3 \cdot 7 \cdot 67 $ | 6.0.5938947.4 | $D_{6}$ (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 *.