Normalized defining polynomial
\( x^{8} - 8x^{6} + 20x^{4} - 32x^{2} + 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(237582155776\)
\(\medspace = 2^{24}\cdot 7^{2}\cdot 17^{2}\)
|
| |
| Root discriminant: | \(26.42\) |
| |
| Galois root discriminant: | $2^{3}7^{1/2}17^{1/2}\approx 87.26969691708571$ | ||
| Ramified primes: |
\(2\), \(7\), \(17\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{15}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{7}{15}$, $\frac{1}{15}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{7}{15}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{15}a^{6}-\frac{2}{3}a^{4}+\frac{5}{3}a^{2}-\frac{7}{15}$, $\frac{7}{15}a^{7}-\frac{11}{3}a^{5}+\frac{26}{3}a^{3}-\frac{199}{15}a$, $\frac{7}{15}a^{7}-\frac{1}{15}a^{6}-\frac{11}{3}a^{5}+\frac{2}{3}a^{4}+\frac{26}{3}a^{3}-\frac{5}{3}a^{2}-\frac{214}{15}a+\frac{37}{15}$, $\frac{7}{15}a^{7}-\frac{1}{5}a^{6}-\frac{11}{3}a^{5}+2a^{4}+\frac{23}{3}a^{3}-5a^{2}-\frac{109}{15}a-\frac{3}{5}$, $\frac{7}{5}a^{7}-\frac{1}{5}a^{6}-11a^{5}+2a^{4}+27a^{3}-5a^{2}-\frac{209}{5}a+\frac{37}{5}$
|
| |
| Regulator: | \( 628.330821225 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 628.330821225 \cdot 2}{2\cdot\sqrt{237582155776}}\cr\approx \mathstrut & 0.814256386358 \end{aligned}\]
Galois group
$C_2^2\wr C_2$ (as 8T18):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_2^2 \wr C_2$ |
| Character table for $C_2^2 \wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.34816.1, 4.2.30464.1, 4.2.14336.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| 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 | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.24c1.61 | $x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $$[2, 3, 4]$$ |
|
\(7\)
| 7.1.2.1a1.1 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 7.1.2.1a1.1 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(17\)
| 17.2.1.0a1.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 17.2.1.0a1.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 17.2.2.2a1.2 | $x^{4} + 32 x^{3} + 262 x^{2} + 96 x + 26$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |