Normalized defining polynomial
\( x^{6} - x^{5} - 5x^{3} - 121x - 29 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[2, 2]$ |
| |
| Discriminant: |
\(67528125\)
\(\medspace = 3^{2}\cdot 5^{5}\cdot 7^{4}\)
|
| |
| Root discriminant: | \(20.18\) |
| |
| Galois root discriminant: | $3^{1/2}5^{5/6}7^{2/3}\approx 24.234514531814607$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\)
|
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{32}a^{4}+\frac{1}{8}a^{3}-\frac{3}{32}a^{2}-\frac{1}{2}a-\frac{11}{32}$, $\frac{1}{96}a^{5}+\frac{1}{96}a^{4}+\frac{17}{96}a^{3}-\frac{7}{96}a^{2}+\frac{37}{96}a+\frac{1}{96}$
| Monogenic: | No | |
| Index: | $4$ | |
| Inessential primes: | $2$ |
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: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{96}a^{5}-\frac{1}{48}a^{4}+\frac{5}{96}a^{3}+\frac{1}{48}a^{2}-\frac{11}{96}a-\frac{31}{48}$, $\frac{1}{48}a^{5}-\frac{13}{96}a^{4}-\frac{13}{48}a^{3}+\frac{31}{96}a^{2}+\frac{157}{48}a+\frac{71}{96}$, $\frac{1}{96}a^{5}-\frac{7}{48}a^{4}+\frac{53}{96}a^{3}-\frac{53}{48}a^{2}+\frac{133}{96}a+\frac{11}{48}$
|
| |
| Regulator: | \( 27.6637780073 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 27.6637780073 \cdot 3}{2\cdot\sqrt{67528125}}\cr\approx \mathstrut & 0.797407162694 \end{aligned}\]
Galois group
| A solvable group of order 12 |
| The 6 conjugacy class representatives for $D_{6}$ |
| Character table for $D_{6}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 3.1.3675.1 |
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: | 3.1.3675.1 $\times$ \(\Q(\sqrt{-15}) \) $\times$ \(\Q\) |
| Degree 6 sibling: | 6.0.202584375.1 |
| 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 | ${\href{/padicField/2.2.0.1}{2} }^{3}$ | R | R | R | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.1.6.5a1.1 | $x^{6} + 5$ | $6$ | $1$ | $5$ | $D_{6}$ | $$[\ ]_{6}^{2}$$ |
|
\(7\)
| 7.2.3.4a1.1 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 169 x + 27$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.3675.3t2.a.a | $2$ | $ 3 \cdot 5^{2} \cdot 7^{2}$ | 3.1.3675.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.3675.6t3.e.a | $2$ | $ 3 \cdot 5^{2} \cdot 7^{2}$ | 6.2.67528125.1 | $D_{6}$ (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 *.