Normalized defining polynomial
\( x^{8} - 4x^{7} + 12x^{6} - 12x^{5} + 12x^{4} + 28x^{3} - 20x^{2} - 12x + 9 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(23592960000\)
\(\medspace = 2^{22}\cdot 3^{2}\cdot 5^{4}\)
|
| |
| Root discriminant: | \(19.80\) |
| |
| Galois root discriminant: | $2^{11/4}3^{1/2}5^{1/2}\approx 26.054222497305222$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{2}, \sqrt{-5})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5409}a^{7}+\frac{2261}{5409}a^{6}-\frac{382}{1803}a^{5}+\frac{206}{1803}a^{4}-\frac{383}{1803}a^{3}-\frac{728}{5409}a^{2}+\frac{805}{5409}a+\frac{160}{1803}$
| 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: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{308}{5409}a^{7}-\frac{1373}{5409}a^{6}+\frac{1342}{1803}a^{5}-\frac{1460}{1803}a^{4}+\frac{1034}{1803}a^{3}+\frac{8363}{5409}a^{2}-\frac{11692}{5409}a-\frac{3007}{1803}$, $\frac{47}{1803}a^{7}-\frac{110}{1803}a^{6}+\frac{76}{601}a^{5}+\frac{66}{601}a^{4}+\frac{29}{601}a^{3}+\frac{1844}{1803}a^{2}-\frac{28}{1803}a-\frac{293}{601}$, $\frac{8600}{5409}a^{7}-\frac{27800}{5409}a^{6}+\frac{26911}{1803}a^{5}-\frac{13370}{1803}a^{4}+\frac{23720}{1803}a^{3}+\frac{284090}{5409}a^{2}+\frac{26525}{5409}a-\frac{32143}{1803}$
|
| |
| Regulator: | \( 93.7211746097 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 93.7211746097 \cdot 2}{2\cdot\sqrt{23592960000}}\cr\approx \mathstrut & 0.950968169726 \end{aligned}\]
Galois group
$C_2\times D_4$ (as 8T9):
| A solvable group of order 16 |
| The 10 conjugacy class representatives for $D_4\times C_2$ |
| Character table for $D_4\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{2}) \), 4.0.38400.4, 4.0.38400.2, \(\Q(\sqrt{2}, \sqrt{-5})\) |
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: | 8.4.8493465600.3, 8.0.13271040000.14, 8.0.53084160000.21 |
| Minimal sibling: | 8.4.8493465600.3 |
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 | R | ${\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.22d1.22 | $x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 14$ | $8$ | $1$ | $22$ | $D_4$ | $$[2, 3, \frac{7}{2}]$$ |
|
\(3\)
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |