Normalized defining polynomial
\( x^{6} - 3x^{5} + 9x^{4} - 13x^{3} + 15x^{2} - 9x + 3 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[0, 3]$ |
| |
| Discriminant: |
\(-465831\)
\(\medspace = -\,3^{8}\cdot 71\)
|
| |
| Root discriminant: | \(8.80\) |
| |
| Galois root discriminant: | $3^{4/3}71^{1/2}\approx 36.45783266912841$ | ||
| Ramified primes: |
\(3\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-71}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | 8.0.166726039041.1$^{4}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
| Monogenic: | Yes | |
| 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 $1$ |
| |
| Relative class number: | $1$ |
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{2}-a+1$, $a^{4}-2a^{3}+6a^{2}-5a+5$
|
| |
| Regulator: | \( 3.39714980258 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 3.39714980258 \cdot 1}{2\cdot\sqrt{465831}}\cr\approx \mathstrut & 0.617319668575 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 6T6):
| A solvable group of order 24 |
| The 8 conjugacy class representatives for $A_4\times C_2$ |
| Character table for $A_4\times C_2$ |
Intermediate fields
| \(\Q(\zeta_{9})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | deg 24 |
| Twin sextic algebra: | 4.4.408321.1 $\times$ \(\Q(\sqrt{-71}) \) |
| Degree 8 sibling: | 8.0.166726039041.1 |
| Degree 12 siblings: | 12.0.5514297181968073041.1, 12.0.1093889542148001.2 |
| 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 | ${\href{/padicField/2.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.3.4a2.1 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $$[2]$$ |
| 3.1.3.4a2.1 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $$[2]$$ | |
|
\(71\)
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 71.2.1.0a1.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 71.1.2.1a1.2 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *24 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.71.2t1.a.a | $1$ | $ 71 $ | \(\Q(\sqrt{-71}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *24 | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| 1.639.6t1.b.a | $1$ | $ 3^{2} \cdot 71 $ | 6.0.2348254071.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.639.6t1.b.b | $1$ | $ 3^{2} \cdot 71 $ | 6.0.2348254071.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| *24 | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| 3.408321.4t4.a.a | $3$ | $ 3^{4} \cdot 71^{2}$ | 4.4.408321.1 | $A_4$ (as 4T4) | $1$ | $3$ | |
| *24 | 3.5751.6t6.a.a | $3$ | $ 3^{4} \cdot 71 $ | 6.0.465831.1 | $A_4\times C_2$ (as 6T6) | $1$ | $-3$ |
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 *.