Normalized defining polynomial
\( x^{8} - x^{7} + 50x^{6} + 71x^{5} + 529x^{4} + 2173x^{3} + 842x^{2} + 5545x + 17623 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(151939915084881\)
\(\medspace = 3^{6}\cdot 7^{6}\cdot 11^{6}\)
|
| |
| Root discriminant: | \(59.25\) |
| |
| Galois root discriminant: | $3^{3/4}7^{3/4}11^{3/4}\approx 59.252814612259755$ | ||
| Ramified primes: |
\(3\), \(7\), \(11\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $Q_8$ |
| |
| This field is Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | 8.0.151939915084881.1$^{8}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{17}a^{6}+\frac{5}{17}a^{5}+\frac{3}{17}a^{4}-\frac{7}{17}a^{3}+\frac{1}{17}a^{2}-\frac{2}{17}a+\frac{5}{17}$, $\frac{1}{36270630857}a^{7}+\frac{374998582}{36270630857}a^{6}+\frac{17017736938}{36270630857}a^{5}+\frac{14057084747}{36270630857}a^{4}-\frac{15124706705}{36270630857}a^{3}+\frac{4190455030}{36270630857}a^{2}-\frac{10103876566}{36270630857}a-\frac{12323908292}{36270630857}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{6}\times C_{6}$, which has order $72$ |
| |
| Narrow class group: | $C_{2}\times C_{6}\times C_{6}$, which has order $72$ |
| |
| Relative class number: | $72$ |
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{5745961}{36270630857}a^{7}-\frac{6527976}{36270630857}a^{6}+\frac{222373728}{36270630857}a^{5}+\frac{298263703}{36270630857}a^{4}+\frac{1372164699}{36270630857}a^{3}+\frac{7555042615}{36270630857}a^{2}-\frac{8494818489}{36270630857}a-\frac{9756958913}{36270630857}$, $\frac{16383577}{36270630857}a^{7}-\frac{10783786}{36270630857}a^{6}+\frac{727410945}{36270630857}a^{5}+\frac{1312883715}{36270630857}a^{4}+\frac{5457923391}{36270630857}a^{3}+\frac{24391519775}{36270630857}a^{2}-\frac{23054803265}{36270630857}a-\frac{99979153583}{36270630857}$, $\frac{5149937}{36270630857}a^{7}+\frac{240974}{36270630857}a^{6}+\frac{89606258}{36270630857}a^{5}+\frac{396628742}{36270630857}a^{4}-\frac{230494035}{36270630857}a^{3}-\frac{38679001}{36270630857}a^{2}+\frac{5709942646}{36270630857}a-\frac{5048433697}{36270630857}$
|
| |
| Regulator: | \( 104.827448547 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 104.827448547 \cdot 72}{2\cdot\sqrt{151939915084881}}\cr\approx \mathstrut & 0.477156725886 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $Q_8$ |
| Character table for $Q_8$ |
Intermediate fields
| \(\Q(\sqrt{33}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{77}) \), \(\Q(\sqrt{21}, \sqrt{33})\) |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.1.0.1}{1} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{8}$ | ${\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 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 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2646 x^{4} + 3240 x^{3} + 2052 x^{2} + 655 x + 109$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
|
\(11\)
| 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 | 1.77.2t1.a.a | $1$ | $ 7 \cdot 11 $ | \(\Q(\sqrt{77}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.33.2t1.a.a | $1$ | $ 3 \cdot 11 $ | \(\Q(\sqrt{33}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *16 | 2.53361.8t5.b.a | $2$ | $ 3^{2} \cdot 7^{2} \cdot 11^{2}$ | 8.0.151939915084881.1 | $Q_8$ (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 *.