Normalized defining polynomial
\( x^{8} - x^{7} - 16x^{6} + 26x^{5} + 73x^{4} - 176x^{3} + 38x^{2} + 55x + 67 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(5489031744\)
\(\medspace = 2^{6}\cdot 3^{6}\cdot 7^{6}\)
|
| |
| Root discriminant: | \(16.50\) |
| |
| Galois root discriminant: | $2^{3/2}3^{3/4}7^{3/4}\approx 27.746580271315977$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}, \sqrt{-7})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{43}a^{6}-\frac{14}{43}a^{5}-\frac{16}{43}a^{4}-\frac{13}{43}a^{3}+\frac{15}{43}a^{2}+\frac{17}{43}a+\frac{11}{43}$, $\frac{1}{9245}a^{7}-\frac{29}{9245}a^{6}+\frac{796}{9245}a^{5}-\frac{3772}{9245}a^{4}+\frac{3994}{9245}a^{3}-\frac{1068}{9245}a^{2}+\frac{2207}{9245}a+\frac{2974}{9245}$
| 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: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( \frac{22}{9245} a^{7} + \frac{7}{9245} a^{6} - \frac{763}{9245} a^{5} - \frac{854}{9245} a^{4} + \frac{5523}{9245} a^{3} + \frac{4669}{9245} a^{2} - \frac{14441}{9245} a - \frac{1437}{9245} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{102}{9245}a^{7}+\frac{267}{9245}a^{6}-\frac{938}{9245}a^{5}-\frac{1829}{9245}a^{4}+\frac{4908}{9245}a^{3}+\frac{4154}{9245}a^{2}-\frac{6656}{9245}a-\frac{3242}{9245}$, $\frac{481}{9245}a^{7}+\frac{671}{9245}a^{6}-\frac{6704}{9245}a^{5}-\frac{5107}{9245}a^{4}+\frac{29974}{9245}a^{3}+\frac{1432}{9245}a^{2}-\frac{2683}{9245}a-\frac{8071}{9245}$, $\frac{40}{1849}a^{7}+\frac{87}{1849}a^{6}-\frac{410}{1849}a^{5}-\frac{724}{1849}a^{4}+\frac{1176}{1849}a^{3}+\frac{1871}{1849}a^{2}+\frac{388}{1849}a-\frac{451}{1849}$
|
| |
| Regulator: | \( 66.8791967631 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 66.8791967631 \cdot 2}{6\cdot\sqrt{5489031744}}\cr\approx \mathstrut & 0.468966041520 \end{aligned}\]
Galois group
| A solvable group of order 16 |
| The 10 conjugacy class representatives for $Q_8:C_2$ |
| Character table for $Q_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 16 |
| Degree 8 siblings: | 8.0.351298031616.5, 8.4.351298031616.1 |
| Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.6a1.2 | $x^{4} + 2 x^{3} + 3 x^{2} + 10 x + 3$ | $2$ | $2$ | $6$ | $C_4$ | $$[3]^{2}$$ |
| 2.4.1.0a1.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(3\)
| 3.2.4.6a1.3 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
|
\(7\)
| 7.1.4.3a1.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 7.1.4.3a1.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *16 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.168.2t1.b.a | $1$ | $ 2^{3} \cdot 3 \cdot 7 $ | \(\Q(\sqrt{-42}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *16 | 1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.56.2t1.a.a | $1$ | $ 2^{3} \cdot 7 $ | \(\Q(\sqrt{14}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *16 | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *16 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *16 | 2.3528.8t11.a.a | $2$ | $ 2^{3} \cdot 3^{2} \cdot 7^{2}$ | 8.0.5489031744.1 | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |
| *16 | 2.3528.8t11.a.b | $2$ | $ 2^{3} \cdot 3^{2} \cdot 7^{2}$ | 8.0.5489031744.1 | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |
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 *.