Normalized defining polynomial
\( x^{8} - 10480x^{4} - 6875928 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-1421768654483998300438528\)
\(\medspace = -\,2^{31}\cdot 131^{7}\)
|
| |
| Root discriminant: | \(1044.97\) |
| |
| Galois root discriminant: | $2^{4}131^{7/8}\approx 1139.5473259618282$ | ||
| Ramified primes: |
\(2\), \(131\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-262}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{362}a^{4}-\frac{86}{181}$, $\frac{1}{3258}a^{5}+\frac{457}{1629}a$, $\frac{1}{58644}a^{6}-\frac{2215}{14661}a^{2}$, $\frac{1}{527796}a^{7}-\frac{19091}{263898}a^{3}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}$, which has order $2$ (assuming GRH) |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{3242859}{181}a^{4}+\frac{2008892417}{181}$, $\frac{28\cdots 25}{9774}a^{6}+\frac{55\cdots 69}{181}a^{4}+\frac{87\cdots 40}{4887}a^{2}+\frac{34\cdots 77}{181}$, $\frac{37\cdots 87}{527796}a^{7}-\frac{11\cdots 10}{1629}a^{6}+\frac{12\cdots 52}{1629}a^{5}-\frac{14\cdots 55}{181}a^{4}+\frac{58\cdots 24}{131949}a^{3}-\frac{74\cdots 72}{1629}a^{2}+\frac{76\cdots 08}{1629}a-\frac{87\cdots 93}{181}$, $\frac{99\cdots 45}{263898}a^{7}+\frac{23\cdots 23}{9774}a^{6}+\frac{12\cdots 62}{1629}a^{5}-\frac{11\cdots 49}{181}a^{4}-\frac{55\cdots 71}{131949}a^{3}-\frac{13\cdots 40}{4887}a^{2}-\frac{13\cdots 20}{1629}a+\frac{12\cdots 39}{181}$
|
| |
| Regulator: | \( 3255933387.41 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 3255933387.41 \cdot 1}{2\cdot\sqrt{1421768654483998300438528}}\cr\approx \mathstrut & 1.35466105126 \end{aligned}\] (assuming GRH)
Galois group
$\SD_{16}$ (as 8T8):
| A solvable group of order 16 |
| The 7 conjugacy class representatives for $QD_{16}$ |
| Character table for $QD_{16}$ |
Intermediate fields
| \(\Q(\sqrt{262}) \), 4.2.4604090368.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 16 |
| 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 | ${\href{/padicField/3.2.0.1}{2} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.31a1.86 | $x^{8} + 16 x^{5} + 26$ | $8$ | $1$ | $31$ | $QD_{16}$ | $$[2, 3, 4, 5]$$ |
|
\(131\)
| 131.1.8.7a1.1 | $x^{8} + 131$ | $8$ | $1$ | $7$ | $QD_{16}$ | $$[\ ]_{8}^{2}$$ |