Normalized defining polynomial
\( x^{8} + 1576x^{6} + 776180x^{4} + 122325968x^{2} + 3012276962 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(3234407759571058688\)
\(\medspace = 2^{31}\cdot 197^{4}\)
|
| |
| Root discriminant: | \(205.93\) |
| |
| Galois root discriminant: | $2^{31/8}197^{1/2}\approx 205.9322413175638$ | ||
| Ramified primes: |
\(2\), \(197\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_8$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(6304=2^{5}\cdot 197\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{6304}(1,·)$, $\chi_{6304}(3939,·)$, $\chi_{6304}(1577,·)$, $\chi_{6304}(5515,·)$, $\chi_{6304}(3153,·)$, $\chi_{6304}(787,·)$, $\chi_{6304}(4729,·)$, $\chi_{6304}(2363,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 8.0.3234407759571058688.4$^{8}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{197}a^{2}$, $\frac{1}{197}a^{3}$, $\frac{1}{38809}a^{4}$, $\frac{1}{38809}a^{5}$, $\frac{1}{7645373}a^{6}$, $\frac{1}{7645373}a^{7}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{73586}$, which has order $73586$ (assuming GRH) |
| |
| Narrow class group: | $C_{73586}$, which has order $73586$ (assuming GRH) |
| |
| Relative class number: | $73586$ (assuming GRH) |
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{38809}a^{4}+\frac{4}{197}a^{2}+1$, $\frac{1}{7645373}a^{6}+\frac{6}{38809}a^{4}+\frac{9}{197}a^{2}+3$, $\frac{1}{197}a^{2}+3$
|
| |
| Regulator: | \( 19.534360053 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 19.534360053 \cdot 73586}{2\cdot\sqrt{3234407759571058688}}\cr\approx \mathstrut & 0.62285431436 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 8 |
| The 8 conjugacy class representatives for $C_8$ |
| Character table for $C_8$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.1.0.1}{1} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.31a1.170 | $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 34$ | $8$ | $1$ | $31$ | $C_8$ | $$[3, 4, 5]$$ |
|
\(197\)
| 197.4.2.4a1.1 | $x^{8} + 32 x^{6} + 248 x^{5} + 260 x^{4} + 3968 x^{3} + 15440 x^{2} + 693 x + 4$ | $2$ | $4$ | $4$ | $C_8$ | $$[\ ]_{2}^{4}$$ |