Normalized defining polynomial
\( x^{8} - 524x^{6} + 63666x^{4} - 1857580x^{2} + 21913811 \)
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{42372567}a^{6}+\frac{4523222}{42372567}a^{4}-\frac{16973857}{42372567}a^{2}-\frac{14649419}{42372567}$, $\frac{1}{17330379903}a^{7}+\frac{1854791981}{17330379903}a^{5}-\frac{4423720825}{17330379903}a^{3}-\frac{8051312960}{17330379903}a$
| 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: |
$\frac{30266684}{455619}a^{6}-\frac{12826588298}{455619}a^{4}+\frac{430478722720}{455619}a^{2}-\frac{5612767073317}{455619}$, $\frac{144341626183154}{17330379903}a^{7}-\frac{6712227926956}{42372567}a^{6}-\frac{23\cdots 56}{17330379903}a^{5}+\frac{10\cdots 81}{42372567}a^{4}+\frac{71\cdots 82}{17330379903}a^{3}-\frac{33\cdots 56}{42372567}a^{2}-\frac{87\cdots 22}{17330379903}a+\frac{40\cdots 82}{42372567}$, $\frac{12\cdots 37}{17330379903}a^{7}-\frac{34\cdots 59}{42372567}a^{6}-\frac{50\cdots 85}{17330379903}a^{5}+\frac{13\cdots 16}{42372567}a^{4}+\frac{16\cdots 73}{17330379903}a^{3}-\frac{46\cdots 23}{42372567}a^{2}-\frac{21\cdots 03}{17330379903}a+\frac{60\cdots 57}{42372567}$, $\frac{36\cdots 07}{5776793301}a^{7}+\frac{50\cdots 13}{42372567}a^{6}-\frac{19\cdots 53}{1925597767}a^{5}-\frac{82\cdots 59}{42372567}a^{4}+\frac{17\cdots 85}{5776793301}a^{3}+\frac{25\cdots 55}{42372567}a^{2}-\frac{72\cdots 99}{1925597767}a-\frac{30\cdots 88}{42372567}$, $\frac{10\cdots 38}{5776793301}a^{7}+\frac{77\cdots 60}{14124189}a^{6}-\frac{41\cdots 85}{5776793301}a^{5}-\frac{30\cdots 80}{14124189}a^{4}+\frac{13\cdots 14}{5776793301}a^{3}+\frac{10\cdots 10}{14124189}a^{2}-\frac{18\cdots 07}{5776793301}a-\frac{13\cdots 01}{14124189}$
|
| |
| Regulator: | \( 5113367053.92 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 5113367053.92 \cdot 1}{2\cdot\sqrt{710884327241999150219264}}\cr\approx \mathstrut & 1.91539003156 \end{aligned}\] (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 | ${\href{/padicField/3.2.0.1}{2} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.30a1.47 | $x^{8} + 8 x^{7} + 8 x^{6} + 16 x^{5} + 26$ | $8$ | $1$ | $30$ | $C_2 \wr C_2\wr C_2$ | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$ |
|
\(131\)
| 131.1.8.7a1.1 | $x^{8} + 131$ | $8$ | $1$ | $7$ | $QD_{16}$ | $$[\ ]_{8}^{2}$$ |