Normalized defining polynomial
\( x^{8} - 2x^{7} - 5x^{6} + 8x^{5} - 6x^{4} + 22x^{3} - 5x^{2} - 28x + 31 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(4405640625\)
\(\medspace = 3^{4}\cdot 5^{6}\cdot 59^{2}\)
|
| |
| Root discriminant: | \(16.05\) |
| |
| Galois root discriminant: | $3^{1/2}5^{3/4}59^{1/2}\approx 44.48505546908213$ | ||
| Ramified primes: |
\(3\), \(5\), \(59\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}$, $\frac{1}{1912}a^{7}-\frac{99}{1912}a^{6}+\frac{19}{956}a^{5}+\frac{73}{956}a^{4}+\frac{43}{478}a^{3}-\frac{205}{956}a^{2}-\frac{387}{1912}a+\frac{227}{1912}$
| 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: |
$\frac{3}{239}a^{7}+\frac{7}{956}a^{6}-\frac{11}{478}a^{5}-\frac{40}{239}a^{4}-\frac{163}{478}a^{3}+\frac{169}{478}a^{2}+\frac{34}{239}a+\frac{95}{956}$, $\frac{45}{1912}a^{7}-\frac{153}{1912}a^{6}-\frac{101}{956}a^{5}+\frac{417}{956}a^{4}+\frac{23}{478}a^{3}+\frac{335}{956}a^{2}-\frac{2119}{1912}a-\frac{3647}{1912}$, $\frac{19}{1912}a^{7}+\frac{31}{1912}a^{6}-\frac{117}{956}a^{5}-\frac{47}{956}a^{4}+\frac{50}{239}a^{3}-\frac{549}{956}a^{2}+\frac{1251}{1912}a+\frac{1445}{1912}$, $\frac{31}{1912}a^{7}-\frac{201}{1912}a^{6}+\frac{111}{956}a^{5}+\frac{351}{956}a^{4}-\frac{170}{239}a^{3}+\frac{815}{956}a^{2}-\frac{4349}{1912}a+\frac{3213}{1912}$, $\frac{105}{1912}a^{7}+\frac{121}{1912}a^{6}-\frac{395}{956}a^{5}-\frac{461}{956}a^{4}-\frac{13}{239}a^{3}-\frac{15}{956}a^{2}+\frac{2385}{1912}a-\frac{2933}{1912}$
|
| |
| Regulator: | \( 48.6572550752 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 48.6572550752 \cdot 1}{2\cdot\sqrt{4405640625}}\cr\approx \mathstrut & 0.231522282226 \end{aligned}\]
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: | 8.0.68160155625.2 |
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 | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.4a1.2 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |
|
\(5\)
| 5.2.4.6a1.2 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |
|
\(59\)
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 59.1.2.1a1.1 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 59.1.2.1a1.1 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |