Normalized defining polynomial
\( x^{8} - 22x^{6} + 29x^{4} + 952x^{2} - 499 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[6, 1]$ |
| |
| Discriminant: |
\(-510976000000\)
\(\medspace = -\,2^{16}\cdot 5^{6}\cdot 499\)
|
| |
| Root discriminant: | \(29.08\) |
| |
| Galois root discriminant: | $2^{5/2}5^{3/4}499^{1/2}\approx 422.52534518485777$ | ||
| Ramified primes: |
\(2\), \(5\), \(499\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-499}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{55651}a^{6}-\frac{541}{1919}a^{4}-\frac{375}{1919}a^{2}-\frac{23785}{55651}$, $\frac{1}{55651}a^{7}-\frac{541}{1919}a^{5}-\frac{375}{1919}a^{3}-\frac{23785}{55651}a$
| 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: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{6}{1919}a^{6}-\frac{103}{1919}a^{4}-\frac{4}{1919}a^{2}+\frac{3134}{1919}$, $\frac{181}{55651}a^{6}-\frac{52}{1919}a^{4}-\frac{710}{1919}a^{2}-\frac{19958}{55651}$, $\frac{71}{55651}a^{6}-\frac{31}{1919}a^{4}+\frac{241}{1919}a^{2}-\frac{19205}{55651}$, $\frac{103}{55651}a^{6}-\frac{72}{1919}a^{4}-\frac{245}{1919}a^{2}-a-\frac{1211}{55651}$, $\frac{547}{55651}a^{7}+\frac{373}{55651}a^{6}-\frac{401}{1919}a^{5}-\frac{298}{1919}a^{4}+\frac{208}{1919}a^{3}+\frac{212}{1919}a^{2}+\frac{512798}{55651}a+\frac{366261}{55651}$, $\frac{309}{55651}a^{7}+\frac{1906}{55651}a^{6}-\frac{216}{1919}a^{5}-\frac{643}{1919}a^{4}-\frac{735}{1919}a^{3}-\frac{4720}{1919}a^{2}+\frac{163320}{55651}a+\frac{188308}{55651}$
|
| |
| Regulator: | \( 813.266917494 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 813.266917494 \cdot 1}{2\cdot\sqrt{510976000000}}\cr\approx \mathstrut & 0.228750842223 \end{aligned}\]
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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\) |
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 | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.16b2.8 | $x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 21 x^{4} + 24 x^{3} + 24 x^{2} + 20 x + 17$ | $4$ | $2$ | $16$ | $C_8:C_2$ | $$[2, 3, 3]^{2}$$ |
|
\(5\)
| 5.2.4.6a1.2 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |
|
\(499\)
| $\Q_{499}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{499}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |