Normalized defining polynomial
\( x^{6} - 3x^{5} + 9x^{4} - 13x^{3} - 45x^{2} + 51x + 71 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(4, 1)$ |
| |
| Discriminant: |
\(-12924583\)
\(\medspace = -\,7^{5}\cdot 769\)
|
| |
| Root discriminant: | \(15.32\) |
| |
| Galois root discriminant: | $7^{5/6}769^{1/2}\approx 140.34971548618705$ | ||
| Ramified primes: |
\(7\), \(769\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-5383}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{9}a^{4}+\frac{1}{9}a^{3}-\frac{1}{9}a^{2}-\frac{4}{9}a-\frac{2}{9}$, $\frac{1}{9}a^{5}+\frac{1}{9}a^{3}+\frac{2}{9}a-\frac{4}{9}$
| 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: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{9}a^{4}-\frac{2}{9}a^{3}+\frac{8}{9}a^{2}-\frac{7}{9}a-\frac{26}{9}$, $\frac{1}{9}a^{4}-\frac{2}{9}a^{3}+\frac{11}{9}a^{2}-\frac{10}{9}a-\frac{29}{9}$, $\frac{1}{9}a^{5}-\frac{1}{9}a^{4}+\frac{2}{3}a^{3}+\frac{1}{9}a^{2}-\frac{17}{3}a-\frac{44}{9}$, $a^{5}-\frac{37}{9}a^{4}+\frac{128}{9}a^{3}-\frac{272}{9}a^{2}-\frac{44}{9}a+\frac{434}{9}$
|
| |
| Regulator: | \( 23.4424922291 \) |
| |
| Unit signature rank: | \( 3 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{1}\cdot 23.4424922291 \cdot 1}{2\cdot\sqrt{12924583}}\cr\approx \mathstrut & 0.327767105673 \end{aligned}\]
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
| \(\Q(\zeta_{7})^+\) |
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: | \(\Q(\sqrt{-5383}) \) $\times$ 4.0.28976689.1 |
| Degree 8 sibling: | 8.0.41142776764733329.1 |
| Degree 12 siblings: | deg 12, deg 12 |
| 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 | ${\href{/padicField/2.3.0.1}{3} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.6.0.1}{6} }$ | R | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.1.6.5a1.6 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |
|
\(769\)
| $\Q_{769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |