Normalized defining polynomial
\( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(64000000\)
\(\medspace = 2^{12}\cdot 5^{6}\)
|
| |
| Root discriminant: | \(9.46\) |
| |
| Galois root discriminant: | $2^{3/2}5^{3/4}\approx 9.457416090031758$ | ||
| Ramified primes: |
\(2\), \(5\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2\times C_4$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(40=2^{3}\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{40}(1,·)$, $\chi_{40}(3,·)$, $\chi_{40}(33,·)$, $\chi_{40}(9,·)$, $\chi_{40}(11,·)$, $\chi_{40}(17,·)$, $\chi_{40}(19,·)$, $\chi_{40}(27,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-10}) \), \(\Q(\zeta_{5})\)$^{2}$, 8.0.64000000.1$^{4}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$
| 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 $1$ |
| |
| Relative class number: | $1$ |
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( \frac{1}{8} a^{6} \)
(order $10$)
|
| |
| Fundamental units: |
$\frac{1}{4}a^{4}+1$, $\frac{1}{8}a^{6}-\frac{1}{2}a^{4}+\frac{1}{2}a^{2}-a-1$, $\frac{1}{2}a^{2}+a+1$
|
| |
| Regulator: | \( 18.013034165 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 18.013034165 \cdot 1}{10\cdot\sqrt{64000000}}\cr\approx \mathstrut & 0.35092665695 \end{aligned}\]
Galois group
$C_2\times C_4$ (as 8T2):
| An abelian group of order 8 |
| The 8 conjugacy class representatives for $C_4\times C_2$ |
| Character table for $C_4\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}, \sqrt{5})\), 4.4.8000.1, \(\Q(\zeta_{5})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.1.0.1}{1} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.2.12a1.1 | $x^{8} + 2 x^{5} + 2 x^{4} + x^{2} + 2 x + 3$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $$[3]^{4}$$ |
|
\(5\)
| 5.2.4.6a1.2 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |