Normalized defining polynomial
\( x^{8} - 46x^{4} - 92x^{3} + 299x^{2} - 46x + 184 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(2298257176725\)
\(\medspace = 3^{3}\cdot 5^{2}\cdot 23^{7}\)
|
| |
| Root discriminant: | \(35.09\) |
| |
| Galois root discriminant: | $3^{3/4}5^{1/2}23^{7/8}\approx 79.22054023965084$ | ||
| Ramified primes: |
\(3\), \(5\), \(23\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{69}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-23}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{10}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{10}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{232440}a^{7}+\frac{4571}{232440}a^{6}-\frac{25559}{232440}a^{5}-\frac{29089}{232440}a^{4}-\frac{679}{15496}a^{3}-\frac{67727}{232440}a^{2}-\frac{42979}{116220}a-\frac{11308}{29055}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ |
| |
| Narrow class group: | $C_{3}$, which has order $3$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{699}{38740}a^{7}-\frac{921}{38740}a^{6}-\frac{6601}{38740}a^{5}+\frac{5289}{38740}a^{4}+\frac{5643}{7748}a^{3}-\frac{39633}{38740}a^{2}+\frac{19919}{19370}a-\frac{12349}{9685}$, $\frac{69181}{232440}a^{7}-\frac{78001}{232440}a^{6}-\frac{49343}{232440}a^{5}-\frac{6713}{46488}a^{4}-\frac{895391}{77480}a^{3}-\frac{895027}{46488}a^{2}+\frac{17405549}{116220}a-\frac{3892054}{29055}$, $\frac{458951}{77480}a^{7}-\frac{603299}{77480}a^{6}-\frac{398969}{77480}a^{5}+\frac{6351561}{77480}a^{4}-\frac{9833971}{15496}a^{3}+\frac{76137383}{77480}a^{2}-\frac{13180829}{38740}a+\frac{5157982}{9685}$
|
| |
| Regulator: | \( 3875.77143566 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 3875.77143566 \cdot 3}{2\cdot\sqrt{2298257176725}}\cr\approx \mathstrut & 5.97681175651 \end{aligned}\]
Galois group
| A solvable group of order 64 |
| The 16 conjugacy class representatives for $(C_4^2 : C_2):C_2$ |
| Character table for $(C_4^2 : C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{-23}) \), 4.0.36501.1 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{3}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }$ | R | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 3.1.4.3a1.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ | |
|
\(5\)
| 5.4.1.0a1.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 5.2.2.2a1.1 | $x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
|
\(23\)
| 23.1.8.7a1.1 | $x^{8} + 23$ | $8$ | $1$ | $7$ | $D_{8}$ | $$[\ ]_{8}^{2}$$ |