Normalized defining polynomial
\( x^{8} + 7x^{4} + 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(40960000\)
\(\medspace = 2^{16}\cdot 5^{4}\)
|
| |
| Root discriminant: | \(8.94\) |
| |
| Galois root discriminant: | $2^{2}5^{1/2}\approx 8.94427190999916$ | ||
| Ramified primes: |
\(2\), \(5\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2^3$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(40=2^{3}\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{40}(1,·)$, $\chi_{40}(39,·)$, $\chi_{40}(9,·)$, $\chi_{40}(11,·)$, $\chi_{40}(19,·)$, $\chi_{40}(21,·)$, $\chi_{40}(29,·)$, $\chi_{40}(31,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-10}) \), 8.0.40960000.1$^{4}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{3}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
\( -\frac{2}{3} a^{7} - \frac{13}{3} a^{3} \)
(order $8$)
|
| |
| Fundamental units: |
$\frac{1}{3}a^{6}+\frac{5}{3}a^{2}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{8}{3}a^{2}+\frac{5}{3}a+1$, $\frac{2}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{13}{3}a^{3}-\frac{5}{3}a+\frac{2}{3}$
|
| |
| Regulator: | \( 12.3400472787 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 12.3400472787 \cdot 1}{8\cdot\sqrt{40960000}}\cr\approx \mathstrut & 0.375635246480 \end{aligned}\]
Galois group
| An abelian group of order 8 |
| The 8 conjugacy class representatives for $C_2^3$ |
| Character table for $C_2^3$ |
Intermediate fields
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 | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.16b1.1 | $x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 21 x^{4} + 20 x^{3} + 20 x^{2} + 12 x + 9$ | $4$ | $2$ | $16$ | $C_2^3$ | $$[2, 3]^{2}$$ |
|
\(5\)
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |