Normalized defining polynomial
\( x^{8} - 22008x^{4} + 693776x^{2} - 4989528 \)
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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{6}a^{3}+\frac{1}{3}a$, $\frac{1}{6}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{6}a^{5}+\frac{1}{3}a$, $\frac{1}{4359564}a^{6}+\frac{57880}{1089891}a^{4}+\frac{162253}{1089891}a^{2}-\frac{31677}{121099}$, $\frac{1}{100269972}a^{7}-\frac{13279}{1089891}a^{5}+\frac{2140991}{50134986}a^{3}+\frac{694917}{2785277}a$
| 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: |
$\frac{170}{121099}a^{6}+\frac{1225}{121099}a^{4}-\frac{3622119}{121099}a^{2}+\frac{75097639}{121099}$, $\frac{6069257}{9115452}a^{7}-\frac{25721}{4444}a^{6}+\frac{5030773}{99081}a^{5}-\frac{999893}{3333}a^{4}-\frac{25967813089}{2278863}a^{3}+\frac{189866210}{3333}a^{2}+\frac{90100003900}{759621}a-\frac{648869137}{1111}$, $\frac{6069257}{9115452}a^{7}+\frac{25721}{4444}a^{6}+\frac{5030773}{99081}a^{5}+\frac{999893}{3333}a^{4}-\frac{25967813089}{2278863}a^{3}-\frac{189866210}{3333}a^{2}+\frac{90100003900}{759621}a+\frac{648869137}{1111}$, $\frac{9917}{303}a^{6}+\frac{3282527}{303}a^{4}-\frac{114323176}{303}a^{2}-\frac{20923297105}{101}$, $\frac{1547890291}{50134986}a^{7}-\frac{317912471}{2179782}a^{6}+\frac{774782638}{1089891}a^{5}-\frac{8077267079}{2179782}a^{4}-\frac{16485129765400}{25067493}a^{3}+\frac{3328986888179}{1089891}a^{2}+\frac{19846271827867}{2785277}a-\frac{4001384875217}{121099}$, $\frac{88\cdots 82}{25067493}a^{7}-\frac{50\cdots 01}{4359564}a^{6}+\frac{42\cdots 05}{1089891}a^{5}-\frac{28\cdots 09}{2179782}a^{4}-\frac{19\cdots 03}{25067493}a^{3}+\frac{27\cdots 02}{1089891}a^{2}+\frac{13\cdots 99}{8355831}a-\frac{64\cdots 69}{121099}$
|
| |
| Regulator: | \( 6478030149.88 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 6478030149.88 \cdot 1}{2\cdot\sqrt{355442163620999575109632}}\cr\approx \mathstrut & 2.18468254960 \end{aligned}\] (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 | ${\href{/padicField/3.2.0.1}{2} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.29a1.101 | $x^{8} + 4 x^{6} + 4 x^{4} + 16 x^{3} + 10$ | $8$ | $1$ | $29$ | $C_2 \wr C_2\wr C_2$ | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$ |
|
\(131\)
| 131.1.8.7a1.1 | $x^{8} + 131$ | $8$ | $1$ | $7$ | $QD_{16}$ | $$[\ ]_{8}^{2}$$ |