Normalized defining polynomial
\( x^{8} - 55x^{4} + 90x^{2} + 205 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(209920000000\)
\(\medspace = 2^{16}\cdot 5^{7}\cdot 41\)
|
| |
| Root discriminant: | \(26.02\) |
| |
| Galois root discriminant: | $2^{5/2}5^{7/8}41^{1/2}\approx 148.10361914290496$ | ||
| Ramified primes: |
\(2\), \(5\), \(41\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{205}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{597}a^{6}+\frac{52}{597}a^{4}-\frac{137}{597}a^{2}+\frac{130}{597}$, $\frac{1}{597}a^{7}+\frac{52}{597}a^{5}-\frac{137}{597}a^{3}+\frac{130}{597}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{4}{199}a^{6}+\frac{9}{199}a^{4}-\frac{150}{199}a^{2}+\frac{122}{199}$, $\frac{5}{597}a^{6}+\frac{61}{597}a^{4}-\frac{88}{597}a^{2}-\frac{1937}{597}$, $\frac{17}{597}a^{6}+\frac{88}{597}a^{4}-\frac{538}{597}a^{2}-\frac{974}{597}$, $\frac{127}{597}a^{7}-\frac{78}{199}a^{6}+\frac{145}{199}a^{5}-\frac{275}{199}a^{4}-\frac{5459}{597}a^{3}+\frac{3323}{199}a^{2}-\frac{2324}{199}a+\frac{4586}{199}$, $\frac{79}{199}a^{7}+\frac{598}{597}a^{6}+\frac{1379}{597}a^{5}+\frac{1145}{199}a^{4}-\frac{1669}{199}a^{3}-\frac{13271}{597}a^{2}-\frac{8194}{597}a-\frac{7386}{199}$
|
| |
| Regulator: | \( 439.551078775 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 439.551078775 \cdot 1}{2\cdot\sqrt{209920000000}}\cr\approx \mathstrut & 0.302992699965 \end{aligned}\]
Galois group
$C_2\wr C_4$ (as 8T27):
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $((C_8 : C_2):C_2):C_2$ |
| Character table for $((C_8 : C_2):C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\) |
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: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | R | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.16b2.8 | $x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 21 x^{4} + 24 x^{3} + 24 x^{2} + 20 x + 17$ | $4$ | $2$ | $16$ | $C_8:C_2$ | $$[2, 3, 3]^{2}$$ |
|
\(5\)
| 5.1.8.7a1.1 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $$[\ ]_{8}^{2}$$ |
|
\(41\)
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 41.1.2.1a1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 41.2.1.0a1.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 41.2.1.0a1.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |