Normalized defining polynomial
\( x^{8} - x^{7} + 7x^{6} - 88x^{5} + 352x^{4} - 780x^{3} + 1251x^{2} - 891x + 729 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(47453100346125\)
\(\medspace = 3^{2}\cdot 5^{3}\cdot 59^{6}\)
|
| |
| Root discriminant: | \(51.23\) |
| |
| Galois root discriminant: | $3^{1/2}5^{1/2}59^{3/4}\approx 82.44885679537175$ | ||
| Ramified primes: |
\(3\), \(5\), \(59\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-59}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{1}{6}a^{3}-\frac{1}{6}a^{2}+\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{36}a^{6}+\frac{1}{18}a^{5}+\frac{1}{9}a^{4}-\frac{1}{9}a^{3}-\frac{1}{18}a^{2}-\frac{1}{3}a-\frac{1}{4}$, $\frac{1}{376812}a^{7}-\frac{365}{188406}a^{6}-\frac{410}{94203}a^{5}-\frac{15142}{94203}a^{4}+\frac{2309}{188406}a^{3}+\frac{7234}{31401}a^{2}+\frac{16471}{41868}a-\frac{341}{1163}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{3}\times C_{6}$, which has order $18$ |
| |
| Narrow class group: | $C_{3}\times C_{6}$, which has order $18$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{734}{94203}a^{7}-\frac{2003}{94203}a^{6}+\frac{10357}{188406}a^{5}-\frac{143197}{188406}a^{4}+\frac{718831}{188406}a^{3}-\frac{593657}{62802}a^{2}+\frac{304201}{20934}a-\frac{26415}{2326}$, $\frac{18701}{376812}a^{7}-\frac{13229}{376812}a^{6}+\frac{15368}{94203}a^{5}-\frac{819541}{188406}a^{4}+\frac{1467497}{94203}a^{3}-\frac{637939}{31401}a^{2}+\frac{789809}{41868}a-\frac{48931}{4652}$, $\frac{757}{376812}a^{7}+\frac{2141}{376812}a^{6}-\frac{3187}{188406}a^{5}-\frac{11596}{94203}a^{4}+\frac{20896}{94203}a^{3}-\frac{55519}{62802}a^{2}+\frac{19795}{41868}a-\frac{3293}{4652}$
|
| |
| Regulator: | \( 893.944303353 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 893.944303353 \cdot 18}{2\cdot\sqrt{47453100346125}}\cr\approx \mathstrut & 1.82028817537 \end{aligned}\]
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
| \(\Q(\sqrt{-59}) \), 4.0.616137.1 |
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: | 8.0.53619322425.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | 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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.4.1.0a1.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(59\)
| 59.2.4.6a1.2 | $x^{8} + 232 x^{7} + 20192 x^{6} + 781840 x^{5} + 11397256 x^{4} + 1563680 x^{3} + 80768 x^{2} + 1856 x + 75$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |