Normalized defining polynomial
\( x^{8} - 84x^{6} + 1764x^{4} - 12348x^{2} + 21609 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(8, 0)$ |
| |
| Discriminant: |
\(29365647704064\)
\(\medspace = 2^{24}\cdot 3^{6}\cdot 7^{4}\)
|
| |
| Root discriminant: | \(48.25\) |
| |
| Galois root discriminant: | $2^{3}3^{3/4}7^{1/2}\approx 48.24807027615272$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $Q_8$ |
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{7}a^{2}$, $\frac{1}{7}a^{3}$, $\frac{1}{147}a^{4}$, $\frac{1}{147}a^{5}$, $\frac{1}{1029}a^{6}$, $\frac{1}{1029}a^{7}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{2}$, which has order $8$ |
|
Unit group
| Rank: | $7$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{1029}a^{6}-\frac{11}{147}a^{4}+\frac{8}{7}a^{2}-3$, $\frac{1}{1029}a^{6}-\frac{10}{147}a^{4}+\frac{5}{7}a^{2}+1$, $\frac{1}{343}a^{6}-\frac{31}{147}a^{4}+\frac{19}{7}a^{2}-6$, $\frac{2}{1029}a^{7}-\frac{22}{147}a^{5}+\frac{1}{147}a^{4}+\frac{17}{7}a^{3}-\frac{4}{7}a^{2}-10a+9$, $\frac{1}{147}a^{4}-\frac{4}{7}a^{2}+a+3$, $\frac{1}{1029}a^{7}+\frac{4}{1029}a^{6}-\frac{10}{147}a^{5}-\frac{41}{147}a^{4}+\frac{5}{7}a^{3}+\frac{24}{7}a^{2}+a-3$, $\frac{4}{1029}a^{6}-\frac{1}{147}a^{5}-\frac{41}{147}a^{4}+\frac{3}{7}a^{3}+\frac{24}{7}a^{2}-3a-9$
|
| |
| Regulator: | \( 6040.95115845 \) |
| |
| Unit signature rank: | \( 6 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 6040.95115845 \cdot 2}{2\cdot\sqrt{29365647704064}}\cr\approx \mathstrut & 0.285381290554 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $Q_8$ |
| Character table for $Q_8$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}, \sqrt{3})\) |
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 | R | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 | $x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 30$ | $8$ | $1$ | $24$ | $Q_8$ | $$[2, 3, 4]$$ |
|
\(3\)
| 3.2.4.6a1.3 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
|
\(7\)
| 7.2.2.2a1.1 | $x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 7.2.2.2a1.1 | $x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
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 | 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *16 | 2.112896.8t5.e.a | $2$ | $ 2^{8} \cdot 3^{2} \cdot 7^{2}$ | 8.8.29365647704064.1 | $Q_8$ (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 *.