Normalized defining polynomial
\( x^{8} - 84x^{6} + 2394x^{4} - 24696x^{2} + 44100 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[8, 0]$ |
| |
| Discriminant: |
\(359729184374784\)
\(\medspace = 2^{22}\cdot 3^{6}\cdot 7^{6}\)
|
| |
| Root discriminant: | \(65.99\) |
| |
| Galois root discriminant: | $2^{11/4}3^{3/4}7^{3/4}\approx 65.99286135128618$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{42}a^{4}$, $\frac{1}{210}a^{5}+\frac{2}{5}a$, $\frac{1}{210}a^{6}+\frac{2}{5}a^{2}$, $\frac{1}{210}a^{7}+\frac{2}{5}a^{3}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| 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: |
$\frac{1}{42}a^{6}-\frac{55}{42}a^{4}+19a^{2}-36$, $\frac{1}{210}a^{6}-\frac{13}{42}a^{4}+\frac{27}{5}a^{2}-12$, $\frac{4}{105}a^{6}-\frac{89}{42}a^{4}+\frac{156}{5}a^{2}-62$, $\frac{2}{105}a^{7}+\frac{8}{105}a^{6}-\frac{131}{105}a^{5}-\frac{29}{6}a^{4}+\frac{118}{5}a^{3}+\frac{422}{5}a^{2}-\frac{504}{5}a-246$, $\frac{1}{42}a^{7}+\frac{41}{210}a^{6}-\frac{41}{35}a^{5}-\frac{151}{14}a^{4}+12a^{3}+\frac{767}{5}a^{2}+\frac{238}{5}a-188$, $\frac{1}{35}a^{7}+\frac{11}{210}a^{6}-\frac{50}{21}a^{5}-\frac{85}{21}a^{4}+\frac{327}{5}a^{3}+\frac{522}{5}a^{2}-592a-897$, $\frac{4}{15}a^{7}+\frac{307}{210}a^{6}-\frac{499}{35}a^{5}-\frac{1639}{21}a^{4}+\frac{1017}{5}a^{3}+\frac{5554}{5}a^{2}-\frac{1958}{5}a-2115$
|
| |
| Regulator: | \( 199628.854108 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 199628.854108 \cdot 1}{2\cdot\sqrt{359729184374784}}\cr\approx \mathstrut & 1.34724148383 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $Q_8$ |
| Character table for $Q_8$ |
Intermediate fields
| \(\Q(\sqrt{21}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{6}, \sqrt{14})\) |
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 | ${\href{/padicField/5.1.0.1}{1} }^{8}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.14 | $x^{8} + 4 x^{7} + 18 x^{6} + 40 x^{5} + 67 x^{4} + 72 x^{3} + 66 x^{2} + 52 x + 19$ | $4$ | $2$ | $22$ | $Q_8$ | $$[3, 4]^{2}$$ |
|
\(3\)
| 3.2.4.6a1.3 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
|
\(7\)
| 7.2.4.6a1.3 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2646 x^{4} + 3240 x^{3} + 2052 x^{2} + 655 x + 109$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
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 | 1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (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 | 1.56.2t1.a.a | $1$ | $ 2^{3} \cdot 7 $ | \(\Q(\sqrt{14}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *16 | 2.112896.8t5.i.a | $2$ | $ 2^{8} \cdot 3^{2} \cdot 7^{2}$ | 8.8.359729184374784.3 | $Q_8$ (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 *.