Normalized defining polynomial
\( x^{8} - 8x^{5} + 2x^{4} + 8x^{2} - 16x - 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-17431527424\)
\(\medspace = -\,2^{24}\cdot 1039\)
|
| |
| Root discriminant: | \(19.06\) |
| |
| Galois root discriminant: | $2^{51/16}1039^{1/2}\approx 293.6573564991579$ | ||
| Ramified primes: |
\(2\), \(1039\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1039}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{1}{7}a^{5}-\frac{1}{7}a^{4}+\frac{2}{7}a^{3}+\frac{2}{7}a^{2}+\frac{1}{7}a+\frac{3}{7}$, $\frac{1}{49}a^{7}+\frac{3}{49}a^{6}+\frac{9}{49}a^{5}+\frac{19}{49}a^{4}+\frac{10}{49}a^{3}-\frac{19}{49}a^{2}-\frac{16}{49}$
| 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: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{8}{49}a^{7}-\frac{4}{49}a^{6}+\frac{2}{49}a^{5}-\frac{65}{49}a^{4}+\frac{24}{49}a^{3}-\frac{12}{49}a^{2}+\frac{10}{7}a-\frac{114}{49}$, $a$, $\frac{2}{49}a^{7}+\frac{13}{49}a^{6}+\frac{11}{49}a^{5}-\frac{18}{49}a^{4}-\frac{113}{49}a^{3}-\frac{73}{49}a^{2}+\frac{1}{7}a+\frac{136}{49}$, $\frac{26}{49}a^{7}-\frac{6}{49}a^{6}-\frac{74}{49}a^{5}-\frac{59}{49}a^{4}+\frac{92}{49}a^{3}+\frac{73}{49}a^{2}-\frac{26}{7}a-\frac{31}{49}$
|
| |
| Regulator: | \( 229.692332043 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 229.692332043 \cdot 1}{2\cdot\sqrt{17431527424}}\cr\approx \mathstrut & 0.863074680407 \end{aligned}\]
Galois group
$S_4\wr C_2$ (as 8T47):
| A solvable group of order 1152 |
| The 20 conjugacy class representatives for $S_4\wr C_2$ |
| Character table for $S_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| 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 | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.24b2.1 | $x^{8} + 4 x^{6} + 8 x + 2$ | $8$ | $1$ | $24$ | $C_2^4:C_6$ | $$[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{3}$$ |
|
\(1039\)
| $\Q_{1039}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |