Normalized defining polynomial
\( x^{8} - x^{7} - 34x^{6} + 29x^{5} + 361x^{4} - 305x^{3} - 1090x^{2} + 1345x - 395 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(8, 0)$ |
| |
| Discriminant: |
\(1340095640625\)
\(\medspace = 3^{6}\cdot 5^{6}\cdot 7^{6}\)
|
| |
| Root discriminant: | \(32.80\) |
| |
| Galois root discriminant: | $3^{3/4}5^{3/4}7^{3/4}\approx 32.801369337662685$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}$, $\frac{1}{205}a^{7}-\frac{13}{205}a^{6}+\frac{81}{205}a^{5}-\frac{18}{41}a^{3}-\frac{9}{41}a^{2}+\frac{13}{41}a-\frac{10}{41}$
| 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: |
$\frac{1}{205}a^{7}+\frac{28}{205}a^{6}-\frac{42}{205}a^{5}-\frac{14}{5}a^{4}+\frac{105}{41}a^{3}+\frac{483}{41}a^{2}-\frac{602}{41}a+\frac{113}{41}$, $\frac{48}{205}a^{7}-\frac{10}{41}a^{6}-\frac{1524}{205}a^{5}+\frac{24}{5}a^{4}+\frac{3031}{41}a^{3}-\frac{1170}{41}a^{2}-\frac{8642}{41}a+\frac{4768}{41}$, $\frac{7}{205}a^{7}+\frac{73}{205}a^{6}-\frac{108}{41}a^{5}-\frac{36}{5}a^{4}+\frac{1719}{41}a^{3}+\frac{1167}{41}a^{2}-\frac{7166}{41}a+\frac{3415}{41}$, $\frac{2}{205}a^{7}-\frac{67}{205}a^{6}+\frac{16}{41}a^{5}+\frac{34}{5}a^{4}-\frac{446}{41}a^{3}-\frac{1248}{41}a^{2}+\frac{2404}{41}a-\frac{922}{41}$, $\frac{53}{205}a^{7}+\frac{8}{205}a^{6}-\frac{1693}{205}a^{5}-\frac{3}{5}a^{4}+\frac{3392}{41}a^{3}-\frac{477}{41}a^{2}-\frac{10381}{41}a+\frac{6030}{41}$, $\frac{18}{205}a^{7}-\frac{29}{205}a^{6}-\frac{592}{205}a^{5}+3a^{4}+\frac{1234}{41}a^{3}-\frac{736}{41}a^{2}-\frac{3743}{41}a+\frac{2157}{41}$, $a^{5}-20a^{3}+80a-47$
|
| |
| Regulator: | \( 1621.61298869 \) |
| |
| Unit signature rank: | \( 6 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 1621.61298869 \cdot 2}{2\cdot\sqrt{1340095640625}}\cr\approx \mathstrut & 0.358607429094 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $Q_8$ |
| Character table for $Q_8$ |
Intermediate fields
| \(\Q(\sqrt{21}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{21})\) |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | R | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.4.6a1.3 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
|
\(5\)
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
|
\(7\)
| 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 | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.105.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 7 $ | \(\Q(\sqrt{105}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *16 | 2.11025.8t5.a.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7^{2}$ | 8.8.1340095640625.1 | $Q_8$ (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 *.