Normalized defining polynomial
\( x^{8} - 24x^{6} - 92x^{4} + 1184x^{2} - 2078 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[6, 1]$ |
| |
| Discriminant: |
\(-2231235510272\)
\(\medspace = -\,2^{31}\cdot 1039\)
|
| |
| Root discriminant: | \(34.96\) |
| |
| Galois root discriminant: | $2^{4}1039^{1/2}\approx 515.7363667611584$ | ||
| Ramified primes: |
\(2\), \(1039\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-2078}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{58807}a^{6}-\frac{27347}{58807}a^{4}+\frac{247}{58807}a^{2}+\frac{15208}{58807}$, $\frac{1}{58807}a^{7}-\frac{27347}{58807}a^{5}+\frac{247}{58807}a^{3}+\frac{15208}{58807}a$
| 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: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{736}{58807}a^{6}-\frac{15398}{58807}a^{4}-\frac{112243}{58807}a^{2}+\frac{549021}{58807}$, $\frac{4}{8401}a^{6}-\frac{175}{8401}a^{4}+\frac{988}{8401}a^{2}+\frac{18827}{8401}$, $\frac{269}{58807}a^{6}-\frac{5468}{58807}a^{4}-\frac{51171}{58807}a^{2}+\frac{268497}{58807}$, $\frac{736}{58807}a^{7}-\frac{622}{58807}a^{6}-\frac{15398}{58807}a^{5}+\frac{14611}{58807}a^{4}-\frac{112243}{58807}a^{3}+\frac{81594}{58807}a^{2}+\frac{549021}{58807}a-\frac{579519}{58807}$, $\frac{14759}{58807}a^{7}+\frac{3384}{8401}a^{6}-\frac{315967}{58807}a^{5}-\frac{72441}{8401}a^{4}-\frac{2176420}{58807}a^{3}-\frac{499911}{8401}a^{2}+\frac{11808760}{58807}a+\frac{2712869}{8401}$, $\frac{11229}{58807}a^{7}-\frac{694}{1897}a^{6}-\frac{224537}{58807}a^{5}+\frac{14509}{1897}a^{4}-\frac{1872190}{58807}a^{3}+\frac{118823}{1897}a^{2}+\frac{6111032}{58807}a-\frac{395917}{1897}$
|
| |
| Regulator: | \( 3546.20045228 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 3546.20045228 \cdot 1}{2\cdot\sqrt{2231235510272}}\cr\approx \mathstrut & 0.477331858028 \end{aligned}\]
Galois group
$C_2\wr C_4$ (as 8T27):
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $((C_8 : C_2):C_2):C_2$ |
| Character table for $((C_8 : C_2):C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\) |
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 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 | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.31a1.166 | $x^{8} + 8 x^{6} + 4 x^{4} + 34$ | $8$ | $1$ | $31$ | $C_8:C_2$ | $$[2, 3, 4, 5]$$ |
|
\(1039\)
| $\Q_{1039}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{1039}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{1039}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{1039}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |