Normalized defining polynomial
\( x^{8} - 524x^{6} + 30916x^{4} - 407672x^{2} - 2266038 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[6, 1]$ |
| |
| Discriminant: |
\(-355442163620999575109632\)
\(\medspace = -\,2^{29}\cdot 131^{7}\)
|
| |
| Root discriminant: | \(878.71\) |
| |
| Galois root discriminant: | $2^{137/32}131^{7/8}\approx 1384.8318794479146$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{24537955}a^{6}+\frac{1243512}{4907591}a^{4}-\frac{4449304}{24537955}a^{2}+\frac{5234012}{24537955}$, $\frac{1}{2282029815}a^{7}-\frac{199967719}{456405963}a^{5}-\frac{470670449}{2282029815}a^{3}-\frac{829056458}{2282029815}a$
| 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: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{10347}{24537955}a^{6}-\frac{1084938}{4907591}a^{4}+\frac{339786462}{24537955}a^{2}-\frac{5593598261}{24537955}$, $\frac{90800052}{59995}a^{6}-\frac{8956776558}{11999}a^{4}+\frac{1456264205412}{59995}a^{2}+\frac{6831873165839}{59995}$, $\frac{1156902384856}{24537955}a^{6}-\frac{15141518967085}{4907591}a^{4}+\frac{10\cdots 76}{24537955}a^{2}+\frac{57\cdots 67}{24537955}$, $\frac{157152862878022}{4907591}a^{7}+\frac{43\cdots 04}{24537955}a^{6}-\frac{77\cdots 32}{4907591}a^{5}-\frac{42\cdots 02}{4907591}a^{4}+\frac{25\cdots 20}{4907591}a^{3}+\frac{69\cdots 14}{24537955}a^{2}+\frac{11\cdots 62}{4907591}a+\frac{32\cdots 43}{24537955}$, $\frac{96\cdots 67}{760676605}a^{7}+\frac{19\cdots 04}{24537955}a^{6}-\frac{93\cdots 95}{152135321}a^{5}-\frac{18\cdots 01}{4907591}a^{4}+\frac{11\cdots 77}{760676605}a^{3}+\frac{22\cdots 24}{24537955}a^{2}+\frac{55\cdots 24}{760676605}a+\frac{11\cdots 83}{24537955}$, $\frac{142560919315849}{2282029815}a^{7}-\frac{32885484130337}{24537955}a^{6}-\frac{18\cdots 93}{456405963}a^{5}+\frac{428803566516164}{4907591}a^{4}+\frac{12\cdots 64}{2282029815}a^{3}-\frac{29\cdots 37}{24537955}a^{2}+\frac{68\cdots 48}{2282029815}a-\frac{16\cdots 39}{24537955}$
|
| |
| Regulator: | \( 5238095916.38 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 5238095916.38 \cdot 1}{2\cdot\sqrt{355442163620999575109632}}\cr\approx \mathstrut & 1.76652106842 \end{aligned}\] (assuming GRH)
Galois group
$C_2\wr D_4$ (as 8T35):
| A solvable group of order 128 |
| The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ |
| Character table for $C_2 \wr C_2\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{262}) \), 4.4.2302045184.1 |
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 32 siblings: | data not computed |
| Minimal sibling: | 8.2.355442163620999575109632.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.29a1.104 | $x^{8} + 20 x^{6} + 16 x^{5} + 4 x^{4} + 16 x^{3} + 10$ | $8$ | $1$ | $29$ | $C_2 \wr C_2\wr C_2$ | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$ |
|
\(131\)
| 131.1.8.7a1.1 | $x^{8} + 131$ | $8$ | $1$ | $7$ | $QD_{16}$ | $$[\ ]_{8}^{2}$$ |