Normalized defining polynomial
\( x^{8} - 8x^{6} + 41x^{4} - 102x^{2} + 61 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(27540127744\)
\(\medspace = 2^{16}\cdot 61\cdot 83^{2}\)
|
| |
| Root discriminant: | \(20.18\) |
| |
| Galois root discriminant: | $2^{5/2}61^{1/2}83^{1/2}\approx 402.512111619017$ | ||
| Ramified primes: |
\(2\), \(61\), \(83\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{61}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{227}a^{6}+\frac{74}{227}a^{4}-\frac{20}{227}a^{2}+\frac{74}{227}$, $\frac{1}{227}a^{7}+\frac{74}{227}a^{5}-\frac{20}{227}a^{3}+\frac{74}{227}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{6}{227}a^{6}-\frac{10}{227}a^{4}+\frac{107}{227}a^{2}-\frac{10}{227}$, $\frac{6}{227}a^{6}-\frac{10}{227}a^{4}+\frac{107}{227}a^{2}+\frac{217}{227}$, $\frac{9}{227}a^{6}-\frac{15}{227}a^{4}+\frac{47}{227}a^{2}+a+\frac{212}{227}$, $\frac{20}{227}a^{7}-\frac{23}{227}a^{6}-\frac{109}{227}a^{5}+\frac{114}{227}a^{4}+\frac{508}{227}a^{3}-\frac{448}{227}a^{2}-\frac{790}{227}a+\frac{341}{227}$, $\frac{22}{227}a^{7}-\frac{25}{227}a^{6}-\frac{188}{227}a^{5}+\frac{193}{227}a^{4}+\frac{922}{227}a^{3}-\frac{1089}{227}a^{2}-\frac{2004}{227}a+\frac{2009}{227}$
|
| |
| Regulator: | \( 220.692492874 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 220.692492874 \cdot 1}{2\cdot\sqrt{27540127744}}\cr\approx \mathstrut & 0.420004853319 \end{aligned}\]
Galois group
$C_2\wr S_4$ (as 8T44):
| A solvable group of order 384 |
| The 20 conjugacy class representatives for $C_2 \wr S_4$ |
| Character table for $C_2 \wr S_4$ |
Intermediate fields
| 4.2.1328.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.16b2.8 | $x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 21 x^{4} + 24 x^{3} + 24 x^{2} + 20 x + 17$ | $4$ | $2$ | $16$ | $C_8:C_2$ | $$[2, 3, 3]^{2}$$ |
|
\(61\)
| 61.1.2.1a1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 61.3.1.0a1.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 61.3.1.0a1.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
|
\(83\)
| 83.2.1.0a1.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 83.2.1.0a1.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 83.2.2.2a1.2 | $x^{4} + 164 x^{3} + 6728 x^{2} + 328 x + 87$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |