Normalized defining polynomial
\( x^{8} + 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(16777216\)
\(\medspace = 2^{24}\)
|
| |
| Root discriminant: | \(8.00\) |
| |
| Galois root discriminant: | $2^{3}\approx 8.0$ | ||
| Ramified primes: |
\(2\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2\times C_4$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(16=2^{4}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{16}(1,·)$, $\chi_{16}(3,·)$, $\chi_{16}(5,·)$, $\chi_{16}(7,·)$, $\chi_{16}(9,·)$, $\chi_{16}(11,·)$, $\chi_{16}(13,·)$, $\chi_{16}(15,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), 4.0.2048.2$^{2}$, \(\Q(\zeta_{16})\)$^{4}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$
| Monogenic: | Yes | |
| 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: |
\( a \)
(order $16$)
|
| |
| Fundamental units: |
$a^{6}+a^{4}+a^{2}$, $a^{7}+a^{4}-a^{2}$, $a^{6}+a^{5}+a^{4}$
|
| |
| Regulator: | \( 19.534360053 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 19.534360053 \cdot 1}{16\cdot\sqrt{16777216}}\cr\approx \mathstrut & 0.46455670329 \end{aligned}\]
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
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{8})\), 4.0.2048.2, \(\Q(\zeta_{16})^+\) |
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 | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.1.0.1}{1} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.24c1.1 | $x^{8} + 2 x^{4} + 4 x^{2} + 8 x + 2$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $$[2, 3, 4]$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *8 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *8 | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *8 | 1.16.4t1.a.a | $1$ | $ 2^{4}$ | \(\Q(\zeta_{16})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| *8 | 1.16.4t1.b.a | $1$ | $ 2^{4}$ | 4.0.2048.2 | $C_4$ (as 4T1) | $0$ | $-1$ |
| *8 | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *8 | 1.16.4t1.a.b | $1$ | $ 2^{4}$ | \(\Q(\zeta_{16})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| *8 | 1.16.4t1.b.b | $1$ | $ 2^{4}$ | 4.0.2048.2 | $C_4$ (as 4T1) | $0$ | $-1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.