Normalized defining polynomial
\( x^{8} - 10x^{6} + 25x^{4} - 50x^{2} - 25 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-16384000000\)
\(\medspace = -\,2^{20}\cdot 5^{6}\)
|
| |
| Root discriminant: | \(18.91\) |
| |
| Galois root discriminant: | $2^{51/16}5^{3/4}\approx 30.46215435643493$ | ||
| Ramified primes: |
\(2\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{85}a^{6}-\frac{4}{85}a^{4}+\frac{7}{17}a^{2}-\frac{2}{17}$, $\frac{1}{85}a^{7}-\frac{4}{85}a^{5}+\frac{7}{17}a^{3}-\frac{2}{17}a$
| 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: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{2}{85}a^{6}-\frac{5}{17}a^{4}+\frac{14}{17}a^{2}-\frac{21}{17}$, $\frac{2}{85}a^{6}-\frac{8}{85}a^{4}-\frac{3}{17}a^{2}-\frac{4}{17}$, $\frac{1}{85}a^{7}+\frac{2}{17}a^{6}+\frac{13}{85}a^{5}-\frac{57}{85}a^{4}-\frac{27}{17}a^{3}-\frac{15}{17}a^{2}-\frac{2}{17}a-\frac{20}{17}$, $\frac{1}{17}a^{7}+\frac{6}{85}a^{6}-\frac{37}{85}a^{5}-\frac{15}{17}a^{4}+\frac{1}{17}a^{3}+\frac{25}{17}a^{2}+\frac{24}{17}a+\frac{22}{17}$
|
| |
| Regulator: | \( 148.655297986 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 148.655297986 \cdot 1}{2\cdot\sqrt{16384000000}}\cr\approx \mathstrut & 0.576155912418 \end{aligned}\]
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
| \(\Q(\sqrt{5}) \), 4.2.1600.1 |
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: | 8.2.4096000000.1 |
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 | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.20a2.20 | $x^{8} + 8 x^{7} + 22 x^{6} + 40 x^{5} + 59 x^{4} + 64 x^{3} + 66 x^{2} + 40 x + 23$ | $4$ | $2$ | $20$ | $(C_4^2 : C_2):C_2$ | $$[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$$ |
|
\(5\)
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.1.4.3a1.4 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |