Normalized defining polynomial
\( x^{8} - 132x^{6} + 4356x^{4} - 30492x^{2} + 27225 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(8, 0)$ |
| |
| Discriminant: |
\(21667237072994304\)
\(\medspace = 2^{24}\cdot 3^{6}\cdot 11^{6}\)
|
| |
| Root discriminant: | \(110.15\) |
| |
| Galois root discriminant: | $2^{3}3^{3/4}11^{3/4}\approx 110.14770224334376$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{33}a^{4}$, $\frac{1}{165}a^{5}+\frac{2}{5}a$, $\frac{1}{134805}a^{6}+\frac{166}{26961}a^{4}+\frac{1427}{4085}a^{2}-\frac{142}{817}$, $\frac{1}{134805}a^{7}+\frac{13}{134805}a^{5}+\frac{1427}{4085}a^{3}+\frac{1741}{4085}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{4}$, which has order $8$ |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{4}$, which has order $16$ |
|
Unit group
| Rank: | $7$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{8}{8987}a^{6}-\frac{2956}{26961}a^{4}+\frac{2385}{817}a^{2}-\frac{2334}{817}$, $\frac{4}{12255}a^{6}-\frac{866}{26961}a^{4}+\frac{1513}{4085}a^{2}+\frac{288}{817}$, $\frac{5}{8987}a^{7}+\frac{124}{134805}a^{6}-\frac{9646}{134805}a^{5}-\frac{3109}{26961}a^{4}+\frac{1797}{817}a^{3}+\frac{13548}{4085}a^{2}-\frac{45897}{4085}a-\frac{10255}{817}$, $\frac{2}{26961}a^{7}+\frac{124}{134805}a^{6}-\frac{1504}{134805}a^{5}-\frac{3109}{26961}a^{4}+\frac{403}{817}a^{3}+\frac{13548}{4085}a^{2}-\frac{26708}{4085}a-\frac{8621}{817}$, $\frac{16}{134805}a^{7}+\frac{52}{134805}a^{6}-\frac{2243}{134805}a^{5}-\frac{1172}{26961}a^{4}+\frac{2407}{4085}a^{3}+\frac{4759}{4085}a^{2}-\frac{17896}{4085}a-\frac{4116}{817}$, $\frac{9}{44935}a^{7}+\frac{28}{44935}a^{6}-\frac{2917}{134805}a^{5}-\frac{1579}{26961}a^{4}+\frac{1764}{4085}a^{3}+\frac{1403}{4085}a^{2}-\frac{8549}{4085}a+\frac{1144}{817}$, $\frac{9}{44935}a^{7}-\frac{28}{44935}a^{6}-\frac{2917}{134805}a^{5}+\frac{1579}{26961}a^{4}+\frac{1764}{4085}a^{3}-\frac{1403}{4085}a^{2}-\frac{8549}{4085}a-\frac{1144}{817}$
|
| |
| Regulator: | \( 66283.652576 \) |
| |
| Unit signature rank: | \( 7 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 66283.652576 \cdot 8}{2\cdot\sqrt{21667237072994304}}\cr\approx \mathstrut & 0.46111008554 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $Q_8$ |
| Character table for $Q_8$ |
Intermediate fields
| \(\Q(\sqrt{66}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{6}, \sqrt{11})\) |
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 | R | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
|
\(11\)
| 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 | 1.264.2t1.a.a | $1$ | $ 2^{3} \cdot 3 \cdot 11 $ | \(\Q(\sqrt{66}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (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 | 2.278784.8t5.e.a | $2$ | $ 2^{8} \cdot 3^{2} \cdot 11^{2}$ | 8.8.21667237072994304.1 | $Q_8$ (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 *.