Normalized defining polynomial
\( x^{8} + 8x^{6} - 156x^{4} - 1120x^{2} + 1186 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(1273457803264\)
\(\medspace = 2^{31}\cdot 593\)
|
| |
| Root discriminant: | \(32.59\) |
| |
| Galois root discriminant: | $2^{4}593^{1/2}\approx 389.6254611803495$ | ||
| Ramified primes: |
\(2\), \(593\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{1186}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{4}+\frac{1}{9}a^{2}-\frac{2}{9}$, $\frac{1}{9}a^{5}+\frac{1}{9}a^{3}-\frac{2}{9}a$, $\frac{1}{2619}a^{6}+\frac{26}{873}a^{4}+\frac{22}{873}a^{2}+\frac{881}{2619}$, $\frac{1}{2619}a^{7}+\frac{26}{873}a^{5}+\frac{22}{873}a^{3}+\frac{881}{2619}a$
| 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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{8}{2619}a^{6}+\frac{14}{873}a^{4}-\frac{103}{291}a^{2}-\frac{1391}{2619}$, $\frac{4}{2619}a^{6}+\frac{7}{873}a^{4}-\frac{100}{291}a^{2}-\frac{259}{2619}$, $\frac{11}{2619}a^{6}-\frac{5}{873}a^{4}-\frac{340}{873}a^{2}-\frac{785}{2619}$, $\frac{781}{2619}a^{7}+\frac{784}{2619}a^{6}+\frac{787}{291}a^{5}+\frac{271}{97}a^{4}-\frac{37720}{873}a^{3}-\frac{36199}{873}a^{2}-\frac{973840}{2619}a-\frac{946753}{2619}$, $\frac{94}{291}a^{7}-\frac{2998}{2619}a^{6}+\frac{1900}{291}a^{5}-\frac{19942}{873}a^{4}+\frac{8338}{291}a^{3}-\frac{9290}{97}a^{2}-\frac{2530}{97}a+\frac{309793}{2619}$
|
| |
| Regulator: | \( 1584.48364931 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 1584.48364931 \cdot 1}{2\cdot\sqrt{1273457803264}}\cr\approx \mathstrut & 0.443450558486 \end{aligned}\]
Galois group
$C_2\wr C_4$ (as 8T27):
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $((C_8 : C_2):C_2):C_2$ |
| Character table for $((C_8 : C_2):C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 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 | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.31a1.166 | $x^{8} + 8 x^{6} + 4 x^{4} + 34$ | $8$ | $1$ | $31$ | $C_8:C_2$ | $$[2, 3, 4, 5]$$ |
|
\(593\)
| $\Q_{593}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{593}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |