Normalized defining polynomial
\( x^{8} - 12x^{6} + 54x^{4} - 72x^{2} + 36 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(3057647616\)
\(\medspace = 2^{22}\cdot 3^{6}\)
|
| |
| Root discriminant: | \(15.33\) |
| |
| Galois root discriminant: | $2^{3}3^{3/4}\approx 18.23605645563822$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{2}, \sqrt{-3})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6}a^{4}$, $\frac{1}{6}a^{5}$, $\frac{1}{30}a^{6}+\frac{1}{15}a^{4}+\frac{2}{5}a^{2}+\frac{1}{5}$, $\frac{1}{30}a^{7}+\frac{1}{15}a^{5}+\frac{2}{5}a^{3}+\frac{1}{5}a$
| 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: | $3$ |
| |
| Torsion generator: |
\( -\frac{1}{15} a^{6} + \frac{7}{10} a^{4} - \frac{14}{5} a^{2} + \frac{13}{5} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{1}{6}a^{4}-a^{2}+1$, $\frac{7}{15}a^{7}-\frac{11}{30}a^{6}-\frac{27}{5}a^{5}+\frac{64}{15}a^{4}+\frac{113}{5}a^{3}-\frac{92}{5}a^{2}-\frac{106}{5}a+\frac{99}{5}$, $\frac{1}{30}a^{7}-\frac{1}{10}a^{6}-\frac{4}{15}a^{5}+\frac{29}{30}a^{4}+\frac{2}{5}a^{3}-\frac{11}{5}a^{2}-\frac{4}{5}a+\frac{12}{5}$
|
| |
| Regulator: | \( 114.024089315 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 114.024089315 \cdot 1}{6\cdot\sqrt{3057647616}}\cr\approx \mathstrut & 0.53563767825 \end{aligned}\]
Galois group
| A solvable group of order 16 |
| The 10 conjugacy class representatives for $Q_8:C_2$ |
| Character table for $Q_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-3})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 16.0.149587343098087735296.16 |
| Degree 8 siblings: | 8.4.12230590464.1, 8.0.12230590464.2 |
| 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 | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.52 | $x^{8} + 12 x^{7} + 34 x^{6} + 64 x^{5} + 79 x^{4} + 72 x^{3} + 46 x^{2} + 36 x + 7$ | $4$ | $2$ | $22$ | $Q_8:C_2$ | $$[2, 3, 4]^{2}$$ |
|
\(3\)
| 3.2.4.6a1.3 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *16 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *16 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *16 | 1.24.2t1.b.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{-6}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *16 | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *16 | 2.2304.8t11.e.a | $2$ | $ 2^{8} \cdot 3^{2}$ | 8.0.3057647616.1 | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |
| *16 | 2.2304.8t11.e.b | $2$ | $ 2^{8} \cdot 3^{2}$ | 8.0.3057647616.1 | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |
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 *.