Normalized defining polynomial
\( x^{8} - 1048x^{4} + 169776 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(177721081810499787554816\)
\(\medspace = 2^{28}\cdot 131^{7}\)
|
| |
| Root discriminant: | \(805.78\) |
| |
| Galois root discriminant: | $2^{59/16}131^{7/8}\approx 917.6149756300749$ | ||
| 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{20}a^{4}-\frac{1}{5}$, $\frac{1}{120}a^{5}-\frac{1}{30}a$, $\frac{1}{360}a^{6}-\frac{1}{90}a^{2}$, $\frac{1}{2160}a^{7}+\frac{89}{540}a^{3}$
| 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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{120}a^{6}+\frac{9}{20}a^{4}-\frac{361}{30}a^{2}-\frac{1184}{5}$, $\frac{193362511}{120}a^{6}-\frac{187667787}{4}a^{4}-\frac{9681260611}{30}a^{2}+9396137570$, $\frac{552230587}{72}a^{6}+\frac{1116548549}{5}a^{4}-\frac{27652522267}{18}a^{2}-\frac{223626777321}{5}$, $\frac{12\cdots 29}{60}a^{7}+\frac{19\cdots 53}{180}a^{6}+\frac{35\cdots 11}{60}a^{5}+\frac{15\cdots 38}{5}a^{4}-\frac{12\cdots 73}{30}a^{3}-\frac{19\cdots 01}{90}a^{2}-\frac{17\cdots 61}{15}a-\frac{31\cdots 72}{5}$, $\frac{34\cdots 27}{360}a^{7}-\frac{14\cdots 83}{40}a^{6}+\frac{27\cdots 91}{20}a^{5}-\frac{10\cdots 73}{20}a^{4}-\frac{36\cdots 41}{45}a^{3}+\frac{30\cdots 63}{10}a^{2}-\frac{57\cdots 26}{5}a+\frac{21\cdots 28}{5}$
|
| |
| Regulator: | \( 2871802133.05 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 2871802133.05 \cdot 1}{2\cdot\sqrt{177721081810499787554816}}\cr\approx \mathstrut & 2.15146736789 \end{aligned}\] (assuming GRH)
Galois group
| A solvable group of order 64 |
| The 16 conjugacy class representatives for $(C_4^2 : C_2):C_2$ |
| Character table for $(C_4^2 : C_2):C_2$ |
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: | 8.4.177721081810499787554816.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.28a1.53 | $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 4 x^{4} + 16 x + 10$ | $8$ | $1$ | $28$ | $(C_4^2 : C_2):C_2$ | $$[2, 2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$$ |
|
\(131\)
| 131.1.8.7a1.1 | $x^{8} + 131$ | $8$ | $1$ | $7$ | $QD_{16}$ | $$[\ ]_{8}^{2}$$ |