Normalized defining polynomial
\( x^{6} - 2x^{5} + 3x^{4} + 2x^{2} - 4x - 2 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[2, 2]$ |
| |
| Discriminant: |
\(2211840\)
\(\medspace = 2^{14}\cdot 3^{3}\cdot 5\)
|
| |
| Root discriminant: | \(11.41\) |
| |
| Galois root discriminant: | $2^{3}3^{1/2}5^{1/2}\approx 30.983866769659336$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{15}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{1}{7}a^{3}-\frac{3}{7}a^{2}+\frac{3}{7}$
| 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: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{1}{7}a^{3}+\frac{4}{7}a^{2}+a+\frac{3}{7}$, $\frac{6}{7}a^{5}-\frac{15}{7}a^{4}+\frac{22}{7}a^{3}-\frac{11}{7}a^{2}+2a-\frac{31}{7}$, $\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{1}{7}a^{3}+\frac{11}{7}a^{2}+\frac{17}{7}$
|
| |
| Regulator: | \( 20.8475644833 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 20.8475644833 \cdot 1}{2\cdot\sqrt{2211840}}\cr\approx \mathstrut & 1.10679758870 \end{aligned}\]
Galois group
$\SOPlus(4,2)$ (as 6T13):
| A solvable group of order 72 |
| The 9 conjugacy class representatives for $C_3^2:D_4$ |
| Character table for $C_3^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{6}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 6.2.6144000.1 |
| Degree 6 sibling: | 6.2.6144000.1 |
| Degree 9 sibling: | 9.1.14155776000.1 |
| Degree 12 siblings: | deg 12, deg 12, deg 12, deg 12, deg 12, deg 12 |
| Degree 18 siblings: | deg 18, deg 18, deg 18 |
| Degree 24 siblings: | deg 24, deg 24 |
| Degree 36 siblings: | deg 36, deg 36, deg 36 |
| 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 | R | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.2 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.1.4.11a1.5 | $x^{4} + 10$ | $4$ | $1$ | $11$ | $D_{4}$ | $$[2, 3, 4]$$ | |
|
\(3\)
| 3.3.2.3a1.2 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.3.1.0a1.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *72 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.60.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 5 $ | \(\Q(\sqrt{15}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *72 | 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.40.2t1.a.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{10}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.3840.4t3.n.a | $2$ | $ 2^{8} \cdot 3 \cdot 5 $ | 4.0.153600.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
| *72 | 4.92160.6t13.b.a | $4$ | $ 2^{11} \cdot 3^{2} \cdot 5 $ | 6.2.2211840.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ |
| 4.9216000.12t34.d.a | $4$ | $ 2^{13} \cdot 3^{2} \cdot 5^{3}$ | 6.2.2211840.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.153600.6t13.b.a | $4$ | $ 2^{11} \cdot 3 \cdot 5^{2}$ | 6.2.2211840.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.5529600.12t34.d.a | $4$ | $ 2^{13} \cdot 3^{3} \cdot 5^{2}$ | 6.2.2211840.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ |
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 *.