Normalized defining polynomial
\( x^{8} - 8x^{6} - 12x^{4} + 2 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(2147483648\)
\(\medspace = 2^{31}\)
|
| |
| Root discriminant: | \(14.67\) |
| |
| Galois root discriminant: | $2^{4}\approx 16.0$ | ||
| Ramified primes: |
\(2\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{17}a^{6}+\frac{5}{17}a^{4}+\frac{2}{17}a^{2}-\frac{8}{17}$, $\frac{1}{17}a^{7}+\frac{5}{17}a^{5}+\frac{2}{17}a^{3}-\frac{8}{17}a$
| 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: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{4}{17}a^{6}-\frac{31}{17}a^{4}-\frac{60}{17}a^{2}+\frac{19}{17}$, $\frac{1}{17}a^{6}-\frac{12}{17}a^{4}+\frac{19}{17}a^{2}+\frac{43}{17}$, $\frac{10}{17}a^{6}-\frac{86}{17}a^{4}-\frac{65}{17}a^{2}+\frac{5}{17}$, $\frac{3}{17}a^{7}-\frac{4}{17}a^{6}-\frac{19}{17}a^{5}+\frac{31}{17}a^{4}-\frac{79}{17}a^{3}+\frac{60}{17}a^{2}-\frac{41}{17}a+\frac{15}{17}$, $\frac{1}{17}a^{7}-\frac{4}{17}a^{6}-\frac{12}{17}a^{5}+\frac{31}{17}a^{4}+\frac{19}{17}a^{3}+\frac{60}{17}a^{2}+\frac{43}{17}a+\frac{15}{17}$
|
| |
| Regulator: | \( 71.6378094121 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 71.6378094121 \cdot 1}{2\cdot\sqrt{2147483648}}\cr\approx \mathstrut & 0.488232952585 \end{aligned}\]
Galois group
$\OD_{16}$ (as 8T7):
| A solvable group of order 16 |
| The 10 conjugacy class representatives for $C_8:C_2$ |
| Character table for $C_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 16.0.18446744073709551616.6 |
| 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 | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.1.0.1}{1} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.31a1.165 | $x^{8} + 8 x^{6} + 4 x^{4} + 2$ | $8$ | $1$ | $31$ | $C_8:C_2$ | $$[2, 3, 4, 5]$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *16 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *16 | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.16.4t1.b.a | $1$ | $ 2^{4}$ | 4.0.2048.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| *16 | 1.16.4t1.a.a | $1$ | $ 2^{4}$ | \(\Q(\zeta_{16})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| *16 | 1.16.4t1.a.b | $1$ | $ 2^{4}$ | \(\Q(\zeta_{16})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| 1.16.4t1.b.b | $1$ | $ 2^{4}$ | 4.0.2048.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| *16 | 2.1024.8t7.a.a | $2$ | $ 2^{10}$ | 8.4.2147483648.1 | $C_8:C_2$ (as 8T7) | $0$ | $0$ |
| *16 | 2.1024.8t7.a.b | $2$ | $ 2^{10}$ | 8.4.2147483648.1 | $C_8:C_2$ (as 8T7) | $0$ | $0$ |
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 *.