Normalized defining polynomial
\( x^{8} + 20436x^{4} - 693776x^{2} + 570636 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(710884327241999150219264\)
\(\medspace = 2^{30}\cdot 131^{7}\)
|
| |
| Root discriminant: | \(958.24\) |
| |
| Galois root discriminant: | $2^{137/32}131^{7/8}\approx 1384.8318794479146$ | ||
| Ramified primes: |
\(2\), \(131\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{131}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{6}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{12}a^{5}-\frac{1}{6}a^{3}-\frac{1}{2}a$, $\frac{1}{11067876}a^{6}-\frac{220721}{5533938}a^{4}+\frac{2250503}{5533938}a^{2}+\frac{125520}{307441}$, $\frac{1}{121746636}a^{7}+\frac{31891}{2766969}a^{5}-\frac{3283435}{60873318}a^{3}+\frac{1606324}{10145553}a$
| 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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1080953}{40101}a^{6}-\frac{357795443}{40101}a^{4}+\frac{10478758382}{40101}a^{2}-\frac{2861520251}{13367}$, $\frac{17\cdots 55}{60873318}a^{7}-\frac{25\cdots 77}{1844646}a^{6}-\frac{10\cdots 15}{5533938}a^{5}+\frac{93\cdots 59}{1844646}a^{4}+\frac{94\cdots 39}{30436659}a^{3}-\frac{13\cdots 34}{922323}a^{2}-\frac{84\cdots 15}{3381851}a+\frac{37\cdots 60}{307441}$, $\frac{17\cdots 89}{60873318}a^{7}-\frac{44\cdots 85}{2766969}a^{6}+\frac{24\cdots 84}{2766969}a^{5}-\frac{13\cdots 46}{2766969}a^{4}+\frac{18\cdots 59}{30436659}a^{3}-\frac{94\cdots 86}{2766969}a^{2}-\frac{17\cdots 14}{3381851}a+\frac{89\cdots 29}{307441}$, $\frac{24\cdots 97}{60873318}a^{7}-\frac{62\cdots 97}{2766969}a^{6}-\frac{61\cdots 35}{2766969}a^{5}+\frac{19\cdots 91}{2766969}a^{4}+\frac{98\cdots 70}{30436659}a^{3}-\frac{55\cdots 01}{2766969}a^{2}-\frac{24\cdots 96}{3381851}a+\frac{51\cdots 19}{307441}$, $\frac{18\cdots 99}{121746636}a^{7}+\frac{37\cdots 71}{2766969}a^{6}+\frac{34\cdots 32}{2766969}a^{5}+\frac{31\cdots 34}{2766969}a^{4}+\frac{18\cdots 73}{60873318}a^{3}+\frac{77\cdots 08}{2766969}a^{2}-\frac{34\cdots 53}{3381851}a-\frac{28\cdots 89}{307441}$
|
| |
| Regulator: | \( 4134637608.1 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 4134637608.1 \cdot 1}{2\cdot\sqrt{710884327241999150219264}}\cr\approx \mathstrut & 1.5487727705 \end{aligned}\] (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
| \(\Q(\sqrt{262}) \), 4.2.4604090368.2 |
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 | ${\href{/padicField/3.2.0.1}{2} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.30a1.46 | $x^{8} + 8 x^{7} + 24 x^{6} + 26$ | $8$ | $1$ | $30$ | $C_2 \wr C_2\wr C_2$ | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$ |
|
\(131\)
| 131.1.8.7a1.1 | $x^{8} + 131$ | $8$ | $1$ | $7$ | $QD_{16}$ | $$[\ ]_{8}^{2}$$ |