Normalized defining polynomial
\( x^{6} - x^{4} - 4x^{3} + 21x^{2} + 2x + 4 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[0, 3]$ |
| |
| Discriminant: |
\(-571787\)
\(\medspace = -\,83^{3}\)
|
| |
| Root discriminant: | \(9.11\) |
| |
| Galois root discriminant: | $83^{1/2}\approx 9.1104335791443$ | ||
| Ramified primes: |
\(83\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-83}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $S_3$ |
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-83}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{12}a^{4}+\frac{1}{12}a^{3}-\frac{1}{12}a^{2}+\frac{1}{3}$, $\frac{1}{36}a^{5}-\frac{1}{36}a^{4}-\frac{1}{4}a^{3}-\frac{1}{9}a^{2}-\frac{1}{18}a+\frac{1}{9}$
| Monogenic: | No | |
| Index: | $4$ | |
| Inessential primes: | $2$ |
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: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{5}{36}a^{5}-\frac{1}{18}a^{4}-\frac{1}{6}a^{3}-\frac{23}{36}a^{2}+\frac{49}{18}a-\frac{10}{9}$, $\frac{1}{12}a^{4}+\frac{1}{12}a^{3}-\frac{1}{12}a^{2}+\frac{1}{3}$
|
| |
| Regulator: | \( 3.24912325959 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 3.24912325959 \cdot 1}{2\cdot\sqrt{571787}}\cr\approx \mathstrut & 0.532915875498 \end{aligned}\]
Galois group
| A solvable group of order 6 |
| The 3 conjugacy class representatives for $S_3$ |
| Character table for $S_3$ |
Intermediate fields
| \(\Q(\sqrt{-83}) \), 3.1.83.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 3.1.83.1 $\times$ \(\Q\) $\times$ \(\Q\) $\times$ \(\Q\) |
| Degree 3 sibling: | 3.1.83.1 |
| Minimal sibling: | 3.1.83.1 |
Multiplicative Galois module structure
| $U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $A$ |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{3}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.1.0.1}{1} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(83\)
| 83.1.2.1a1.1 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 83.1.2.1a1.1 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 83.1.2.1a1.1 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |