Normalized defining polynomial
\( x^{8} + 4x^{6} - 6x^{4} - 72x^{2} + 324 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(1078141648896\)
\(\medspace = 2^{22}\cdot 3^{2}\cdot 13^{4}\)
|
| |
| Root discriminant: | \(31.92\) |
| |
| Galois root discriminant: | $2^{11/4}3^{1/2}13^{1/2}\approx 42.011171440958684$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
|
| |
| 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: | \(\Q(\sqrt{-2}, \sqrt{13})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{18}a^{5}-\frac{1}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{162}a^{6}-\frac{11}{162}a^{4}+\frac{13}{27}a^{2}+\frac{1}{3}$, $\frac{1}{162}a^{7}-\frac{1}{81}a^{5}+\frac{10}{27}a^{3}-\frac{1}{3}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ |
| |
| Narrow class group: | $C_{3}$, which has order $3$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{81}a^{6}+\frac{5}{162}a^{4}-\frac{10}{27}a^{2}-\frac{7}{3}$, $\frac{1}{81}a^{7}+\frac{4}{81}a^{6}+\frac{16}{81}a^{5}+\frac{10}{81}a^{4}+\frac{8}{27}a^{3}+\frac{14}{27}a^{2}-\frac{10}{3}a-\frac{19}{3}$, $\frac{4}{81}a^{7}-\frac{5}{81}a^{6}+\frac{19}{81}a^{5}-\frac{26}{81}a^{4}+\frac{8}{27}a^{3}-\frac{22}{27}a^{2}-\frac{8}{3}a+\frac{23}{3}$
|
| |
| Regulator: | \( 270.995731769 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 270.995731769 \cdot 3}{2\cdot\sqrt{1078141648896}}\cr\approx \mathstrut & 0.610148111726 \end{aligned}\]
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
| \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-2}, \sqrt{13})\) |
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: | 8.4.9703274840064.4, 8.0.57415827456.5 |
| Minimal sibling: | 8.0.57415827456.5 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.10 | $x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 19 x^{4} + 16 x^{3} + 18 x^{2} + 28 x + 11$ | $4$ | $2$ | $22$ | $Q_8$ | $$[3, 4]^{2}$$ |
|
\(3\)
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(13\)
| 13.4.2.4a1.2 | $x^{8} + 6 x^{6} + 24 x^{5} + 13 x^{4} + 72 x^{3} + 156 x^{2} + 48 x + 17$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *16 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *16 | 1.104.2t1.b.a | $1$ | $ 2^{3} \cdot 13 $ | \(\Q(\sqrt{-26}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.39.2t1.a.a | $1$ | $ 3 \cdot 13 $ | \(\Q(\sqrt{-39}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *16 | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *16 | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.312.2t1.a.a | $1$ | $ 2^{3} \cdot 3 \cdot 13 $ | \(\Q(\sqrt{78}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *16 | 2.9984.8t11.i.a | $2$ | $ 2^{8} \cdot 3 \cdot 13 $ | 8.0.1078141648896.7 | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |
| *16 | 2.9984.8t11.i.b | $2$ | $ 2^{8} \cdot 3 \cdot 13 $ | 8.0.1078141648896.7 | $Q_8:C_2$ (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 *.