Normalized defining polynomial
\( x^{8} - 3x^{7} + 17x^{6} - 30x^{5} + 91x^{4} - 90x^{3} + 228x^{2} - 54x + 261 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(41463140625\)
\(\medspace = 3^{4}\cdot 5^{6}\cdot 181^{2}\)
|
| |
| Root discriminant: | \(21.24\) |
| |
| Galois root discriminant: | $3^{1/2}5^{3/4}181^{1/2}\approx 77.91613798765907$ | ||
| Ramified primes: |
\(3\), \(5\), \(181\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | 8.0.6037217555625.1$^{4}$, 8.0.1358373950015625.1$^{4}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{15}a^{6}+\frac{1}{15}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{2265}a^{7}+\frac{7}{2265}a^{6}-\frac{64}{2265}a^{5}+\frac{17}{453}a^{4}+\frac{158}{453}a^{3}-\frac{316}{755}a^{2}+\frac{238}{755}a-\frac{356}{755}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{6}$, which has order $12$ |
| |
| Narrow class group: | $C_{2}\times C_{6}$, which has order $12$ |
| |
| Relative class number: | $12$ |
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{9}{755}a^{7}-\frac{113}{2265}a^{6}+\frac{179}{755}a^{5}-\frac{1027}{2265}a^{4}+\frac{768}{755}a^{3}-\frac{1889}{2265}a^{2}+\frac{198}{151}a+\frac{354}{755}$, $\frac{56}{2265}a^{7}-\frac{212}{2265}a^{6}+\frac{946}{2265}a^{5}-\frac{628}{755}a^{4}+\frac{3923}{2265}a^{3}-\frac{3409}{2265}a^{2}+\frac{1701}{755}a+\frac{751}{755}$, $\frac{7}{2265}a^{7}-\frac{34}{755}a^{6}+\frac{307}{2265}a^{5}-\frac{1066}{2265}a^{4}+\frac{434}{755}a^{3}-\frac{542}{453}a^{2}+\frac{458}{755}a-\frac{2039}{755}$
|
| |
| Regulator: | \( 9.32364155459 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 9.32364155459 \cdot 12}{2\cdot\sqrt{41463140625}}\cr\approx \mathstrut & 0.428178834132 \end{aligned}\]
Galois group
$C_2^3:C_4$ (as 8T20):
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $C_2^3: C_4$ |
| Character table for $C_2^3: C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Minimal sibling: | 8.0.6037217555625.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.1.4.3a1.4 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.1.4.3a1.4 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
|
\(181\)
| 181.1.2.1a1.2 | $x^{2} + 362$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 181.2.1.0a1.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 181.2.1.0a1.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 181.1.2.1a1.2 | $x^{2} + 362$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |