Normalized defining polynomial
\( x^{8} + 4x^{6} - 16x^{5} + 12x^{4} - 8x^{3} + 8x - 2 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(12901679104\)
\(\medspace = 2^{24}\cdot 769\)
|
| |
| Root discriminant: | \(18.36\) |
| |
| Galois root discriminant: | $2^{51/16}769^{1/2}\approx 252.6366076436748$ | ||
| Ramified primes: |
\(2\), \(769\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{769}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{769}a^{7}-\frac{232}{769}a^{6}-\frac{2}{769}a^{5}-\frac{321}{769}a^{4}-\frac{109}{769}a^{3}-\frac{97}{769}a^{2}+\frac{203}{769}a-\frac{179}{769}$
| 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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{258}{769}a^{7}+\frac{126}{769}a^{6}+\frac{1022}{769}a^{5}-\frac{3611}{769}a^{4}+\frac{1100}{769}a^{3}-\frac{418}{769}a^{2}-\frac{1456}{769}a+\frac{727}{769}$, $\frac{17}{769}a^{7}-\frac{99}{769}a^{6}-\frac{34}{769}a^{5}-\frac{843}{769}a^{4}+\frac{1223}{769}a^{3}-\frac{880}{769}a^{2}+\frac{1144}{769}a+\frac{33}{769}$, $\frac{132}{769}a^{7}+\frac{136}{769}a^{6}+\frac{505}{769}a^{5}-\frac{1615}{769}a^{4}-\frac{546}{769}a^{3}+\frac{1038}{769}a^{2}-\frac{119}{769}a+\frac{211}{769}$, $\frac{401}{769}a^{7}+\frac{17}{769}a^{6}+\frac{1505}{769}a^{5}-\frac{6450}{769}a^{4}+\frac{3969}{769}a^{3}-\frac{1985}{769}a^{2}-\frac{111}{769}a+\frac{3583}{769}$, $\frac{666}{769}a^{7}+\frac{57}{769}a^{6}+\frac{2513}{769}a^{5}-\frac{10770}{769}a^{4}+\frac{5844}{769}a^{3}-\frac{4620}{769}a^{2}-\frac{915}{769}a+\frac{6133}{769}$
|
| |
| Regulator: | \( 162.054891788 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 162.054891788 \cdot 1}{2\cdot\sqrt{12901679104}}\cr\approx \mathstrut & 0.450597472163 \end{aligned}\]
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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 | $x^{8} + 4 x^{6} + 8 x + 2$ | $8$ | $1$ | $24$ | $C_2^4:C_6$ | $$[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{3}$$ |
|
\(769\)
| $\Q_{769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |