Normalized defining polynomial
\( x^{8} + 44x^{6} + 308x^{4} + 484x^{2} + 121 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(29721861554176\)
\(\medspace = 2^{24}\cdot 11^{6}\)
|
| |
| Root discriminant: | \(48.32\) |
| |
| Galois root discriminant: | $2^{3}11^{3/4}\approx 48.32084283629789$ | ||
| Ramified primes: |
\(2\), \(11\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $Q_8$ |
| |
| This field is Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | 8.0.29721861554176.1$^{8}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{33}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{33}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{231}a^{6}-\frac{1}{3}a^{2}+\frac{2}{21}$, $\frac{1}{231}a^{7}-\frac{1}{3}a^{3}+\frac{2}{21}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}\times C_{6}$, which has order $18$ |
| |
| Narrow class group: | $C_{3}\times C_{6}$, which has order $18$ |
| |
| Relative class number: | $18$ |
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{77}a^{6}+\frac{6}{11}a^{4}+3a^{2}+\frac{23}{7}$, $\frac{1}{231}a^{6}+\frac{5}{33}a^{4}-\frac{5}{21}$, $\frac{9}{77}a^{6}+\frac{53}{11}a^{4}+23a^{2}+\frac{46}{7}$
|
| |
| Regulator: | \( 126.131828435 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 126.131828435 \cdot 18}{2\cdot\sqrt{29721861554176}}\cr\approx \mathstrut & 0.324525382774 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $Q_8$ |
| Character table for $Q_8$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{2}, \sqrt{11})\) |
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 | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.24c1.57 | $x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 30$ | $8$ | $1$ | $24$ | $Q_8$ | $$[2, 3, 4]$$ |
|
\(11\)
| 11.2.4.6a1.3 | $x^{8} + 28 x^{7} + 302 x^{6} + 1540 x^{5} + 3601 x^{4} + 3080 x^{3} + 1208 x^{2} + 268 x + 115$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
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.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.88.2t1.a.a | $1$ | $ 2^{3} \cdot 11 $ | \(\Q(\sqrt{22}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.44.2t1.a.a | $1$ | $ 2^{2} \cdot 11 $ | \(\Q(\sqrt{11}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *16 | 2.30976.8t5.b.a | $2$ | $ 2^{8} \cdot 11^{2}$ | 8.0.29721861554176.1 | $Q_8$ (as 8T5) | $-1$ | $-2$ |
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 *.