Normalized defining polynomial
\( x^{8} - 6x^{4} - 3 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-573308928\)
\(\medspace = -\,2^{18}\cdot 3^{7}\)
|
| |
| Root discriminant: | \(12.44\) |
| |
| Galois root discriminant: | $2^{9/4}3^{7/8}\approx 12.43937579553693$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a$
| 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: |
$\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{5}{4}a^{2}+\frac{1}{4}$, $\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{5}{4}a^{3}-\frac{5}{4}a^{2}-\frac{5}{4}a-\frac{1}{4}$
|
| |
| Regulator: | \( 31.8596637081 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 31.8596637081 \cdot 1}{2\cdot\sqrt{573308928}}\cr\approx \mathstrut & 0.660110187945 \end{aligned}\]
Galois group
$\SD_{16}$ (as 8T8):
| A solvable group of order 16 |
| The 7 conjugacy class representatives for $QD_{16}$ |
| Character table for $QD_{16}$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), 4.2.1728.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 16.0.328683126924509184.1 |
| 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 | R | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{8}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.18b1.12 | $x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 6$ | $8$ | $1$ | $18$ | $QD_{16}$ | $$[2, 2, 3]^{2}$$ |
|
\(3\)
| 3.1.8.7a1.2 | $x^{8} + 6$ | $8$ | $1$ | $7$ | $QD_{16}$ | $$[\ ]_{8}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *16 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *16 | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *16 | 2.144.4t3.c.a | $2$ | $ 2^{4} \cdot 3^{2}$ | 4.2.1728.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| *16 | 2.576.8t8.a.a | $2$ | $ 2^{6} \cdot 3^{2}$ | 8.2.573308928.1 | $QD_{16}$ (as 8T8) | $0$ | $0$ |
| *16 | 2.576.8t8.a.b | $2$ | $ 2^{6} \cdot 3^{2}$ | 8.2.573308928.1 | $QD_{16}$ (as 8T8) | $0$ | $0$ |
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 *.