Normalized defining polynomial
\( x^{8} - 24x^{4} + 169 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(2835349504\)
\(\medspace = 2^{24}\cdot 13^{2}\)
|
| |
| Root discriminant: | \(15.19\) |
| |
| Galois root discriminant: | $2^{3}13^{1/2}\approx 28.844410203711913$ | ||
| Ramified primes: |
\(2\), \(13\)
|
| |
| 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: | \(\Q(\zeta_{8})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}-\frac{2}{5}$, $\frac{1}{5}a^{5}-\frac{2}{5}a$, $\frac{1}{65}a^{6}+\frac{28}{65}a^{2}$, $\frac{1}{65}a^{7}+\frac{28}{65}a^{3}$
| 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: |
\( \frac{2}{65} a^{6} - \frac{9}{65} a^{2} \)
(order $8$)
|
| |
| Fundamental units: |
$\frac{2}{65}a^{6}+\frac{1}{5}a^{4}-\frac{9}{65}a^{2}-\frac{7}{5}$, $\frac{3}{65}a^{7}-\frac{1}{65}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{46}{65}a^{3}-\frac{28}{65}a^{2}+\frac{12}{5}a-\frac{7}{5}$, $\frac{2}{65}a^{7}+\frac{1}{65}a^{6}+\frac{1}{5}a^{4}-\frac{9}{65}a^{3}+\frac{28}{65}a^{2}+a-\frac{7}{5}$
|
| |
| Regulator: | \( 83.8010606528 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 83.8010606528 \cdot 1}{8\cdot\sqrt{2835349504}}\cr\approx \mathstrut & 0.306602506981 \end{aligned}\]
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
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{8})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 16 |
| Degree 8 siblings: | 8.0.479174066176.4, 8.4.119793516544.2 |
| 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 | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.24c1.3 | $x^{8} + 8 x^{7} + 2 x^{4} + 4 x^{2} + 8 x + 2$ | $8$ | $1$ | $24$ | $Q_8:C_2$ | $$[2, 3, 4]^{2}$$ |
|
\(13\)
| 13.2.2.2a1.1 | $x^{4} + 24 x^{3} + 148 x^{2} + 61 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 13.4.1.0a1.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *16 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.104.2t1.b.a | $1$ | $ 2^{3} \cdot 13 $ | \(\Q(\sqrt{-26}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.52.2t1.a.a | $1$ | $ 2^{2} \cdot 13 $ | \(\Q(\sqrt{-13}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *16 | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.104.2t1.a.a | $1$ | $ 2^{3} \cdot 13 $ | \(\Q(\sqrt{26}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *16 | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *16 | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *16 | 2.3328.8t11.e.a | $2$ | $ 2^{8} \cdot 13 $ | 8.0.2835349504.2 | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |
| *16 | 2.3328.8t11.e.b | $2$ | $ 2^{8} \cdot 13 $ | 8.0.2835349504.2 | $Q_8:C_2$ (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 *.