Normalized defining polynomial
\( x^{6} - x^{5} + 5x^{4} + 6x^{3} + 7x^{2} + 2x - 1 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[2, 2]$ |
| |
| Discriminant: |
\(946125\)
\(\medspace = 3^{2}\cdot 5^{3}\cdot 29^{2}\)
|
| |
| Root discriminant: | \(9.91\) |
| |
| Galois root discriminant: | $3^{1/2}5^{1/2}29^{1/2}\approx 20.85665361461421$ | ||
| Ramified primes: |
\(3\), \(5\), \(29\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{23}a^{5}+\frac{7}{23}a^{4}-\frac{8}{23}a^{3}+\frac{11}{23}a^{2}+\frac{3}{23}a+\frac{3}{23}$
| 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: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{8}{23}a^{5}-\frac{13}{23}a^{4}+\frac{51}{23}a^{3}+\frac{19}{23}a^{2}+\frac{47}{23}a+\frac{1}{23}$, $\frac{5}{23}a^{5}-\frac{11}{23}a^{4}+\frac{29}{23}a^{3}+\frac{9}{23}a^{2}-\frac{8}{23}a-\frac{8}{23}$, $\frac{2}{23}a^{5}-\frac{9}{23}a^{4}+\frac{30}{23}a^{3}-\frac{47}{23}a^{2}+\frac{52}{23}a-\frac{17}{23}$
|
| |
| Regulator: | \( 4.17502696008 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 4.17502696008 \cdot 1}{2\cdot\sqrt{946125}}\cr\approx \mathstrut & 0.338902488290 \end{aligned}\]
Galois group
| A solvable group of order 12 |
| The 6 conjugacy class representatives for $D_{6}$ |
| Character table for $D_{6}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 3.1.87.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | 12.0.6775409390765625.3 |
| Twin sextic algebra: | 3.1.87.1 $\times$ \(\Q(\sqrt{-435}) \) $\times$ \(\Q\) |
| Degree 6 sibling: | 6.0.82312875.1 |
| 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 | R | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(29\)
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 29.1.2.1a1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 29.1.2.1a1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *12 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.87.2t1.a.a | $1$ | $ 3 \cdot 29 $ | \(\Q(\sqrt{-87}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *12 | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.435.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 29 $ | \(\Q(\sqrt{-435}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *12 | 2.87.3t2.a.a | $2$ | $ 3 \cdot 29 $ | 3.1.87.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| *12 | 2.2175.6t3.d.a | $2$ | $ 3 \cdot 5^{2} \cdot 29 $ | 6.2.946125.1 | $D_{6}$ (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 *.