Normalized defining polynomial
\( x^{8} - 4x^{7} + 4x^{6} - 2x^{5} + 5x^{4} - 2x^{3} + 4x^{2} - 4x + 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(136048896\)
\(\medspace = 2^{8}\cdot 3^{12}\)
|
| |
| Root discriminant: | \(10.39\) |
| |
| Galois root discriminant: | $2^{7/6}3^{11/6}\approx 16.82379477331321$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$
| 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: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -\frac{8}{3} a^{7} + \frac{28}{3} a^{6} - 6 a^{5} + \frac{7}{3} a^{4} - \frac{38}{3} a^{3} - \frac{32}{3} a + \frac{19}{3} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{4}{3}a^{7}-\frac{14}{3}a^{6}+3a^{5}-\frac{2}{3}a^{4}+\frac{16}{3}a^{3}+\frac{13}{3}a-\frac{8}{3}$, $7a^{7}-25a^{6}+17a^{5}-6a^{4}+32a^{3}+28a-15$, $\frac{20}{3}a^{7}-\frac{70}{3}a^{6}+15a^{5}-\frac{16}{3}a^{4}+\frac{89}{3}a^{3}+2a^{2}+\frac{80}{3}a-\frac{40}{3}$
|
| |
| Regulator: | \( 28.1796625828 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 28.1796625828 \cdot 1}{6\cdot\sqrt{136048896}}\cr\approx \mathstrut & 0.627561801060 \end{aligned}\]
Galois group
| A solvable group of order 24 |
| The 5 conjugacy class representatives for $S_4$ |
| Character table for $S_4$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 4.2.3888.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 24 |
| Degree 4 sibling: | 4.2.3888.1 |
| Degree 6 siblings: | 6.2.3779136.3, 6.0.11337408.4 |
| Degree 12 siblings: | 12.2.171382426877952.3, 12.0.128536820158464.9 |
| Minimal sibling: | 4.2.3888.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.8a1.1 | $x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 19 x^{4} + 16 x^{3} + 12 x^{2} + 6 x + 5$ | $4$ | $2$ | $8$ | $S_4$ | $$[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$$ |
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.6.11a1.5 | $x^{6} + 18 x^{2} + 3$ | $6$ | $1$ | $11$ | $S_3$ | $$[\frac{5}{2}]_{2}$$ |