Normalized defining polynomial
\( x^{6} + 8x^{4} - 20x^{3} + 21x^{2} - 70x + 105 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(0, 3)$ |
| |
| Discriminant: |
\(-31752000\)
\(\medspace = -\,2^{6}\cdot 3^{4}\cdot 5^{3}\cdot 7^{2}\)
|
| |
| Root discriminant: | \(17.79\) |
| |
| Galois root discriminant: | $2\cdot 3^{4/3}5^{1/2}7^{2/3}\approx 70.80686465272596$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-5}) \) | ||
| $\Aut(K/\Q)$: | $C_3$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-5}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{645}a^{5}-\frac{286}{645}a^{4}-\frac{37}{215}a^{3}+\frac{121}{645}a^{2}+\frac{49}{129}a+\frac{11}{43}$
| Monogenic: | No | |
| Index: | $3$ | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | $C_{6}$, which has order $6$ |
| |
| Narrow class group: | $C_{6}$, which has order $6$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{2}{645}a^{5}+\frac{73}{645}a^{4}-\frac{74}{215}a^{3}+\frac{242}{645}a^{2}-\frac{418}{129}a+\frac{237}{43}$, $\frac{18}{215}a^{5}+\frac{12}{215}a^{4}+\frac{152}{215}a^{3}-\frac{187}{215}a^{2}+\frac{22}{43}a-\frac{223}{43}$
|
| |
| Regulator: | \( 14.9985936044 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 14.9985936044 \cdot 6}{2\cdot\sqrt{31752000}}\cr\approx \mathstrut & 1.98073262357 \end{aligned}\]
Galois group
$C_3\times S_3$ (as 6T5):
| A solvable group of order 18 |
| The 9 conjugacy class representatives for $S_3\times C_3$ |
| Character table for $S_3\times C_3$ |
Intermediate fields
| \(\Q(\sqrt{-5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | 18.0.2001504424181159874396672000000000.8 |
| Twin sextic algebra: | 3.3.3969.2 $\times$ 3.1.79380.2 |
| Degree 9 sibling: | 9.3.500188017672000.19 |
| 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 | R | ${\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.6a1.2 | $x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $$[2]^{3}$$ |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 3.1.3.4a2.2 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $$[2]$$ | |
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(7\)
| 7.1.3.2a1.1 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 7.3.1.0a1.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *18 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *18 | 1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.63.3t1.a.a | $1$ | $ 3^{2} \cdot 7 $ | 3.3.3969.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.1260.6t1.b.a | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 $ | 6.0.126023688000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.1260.6t1.b.b | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 $ | 6.0.126023688000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.63.3t1.a.b | $1$ | $ 3^{2} \cdot 7 $ | 3.3.3969.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 2.79380.3t2.b.a | $2$ | $ 2^{2} \cdot 3^{4} \cdot 5 \cdot 7^{2}$ | 3.1.79380.2 | $S_3$ (as 3T2) | $1$ | $0$ | |
| *18 | 2.1260.6t5.d.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 $ | 6.0.31752000.5 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| *18 | 2.1260.6t5.d.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 $ | 6.0.31752000.5 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
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 *.