Normalized defining polynomial
\( x^{8} - 4x^{7} - 14x^{6} + 17x^{5} + 43x^{4} + 323x^{3} + 310x^{2} - 1135x - 413 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-321078613651227\) \(\medspace = -\,3^{7}\cdot 619^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(65.06\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $3^{7/8}619^{1/2}\approx 65.06185214674588$ | ||
Ramified primes: | \(3\), \(619\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{14371412}a^{7}-\frac{136173}{3592853}a^{6}+\frac{173901}{14371412}a^{5}+\frac{1890737}{7185706}a^{4}-\frac{5556523}{14371412}a^{3}-\frac{4852817}{14371412}a^{2}+\frac{5641453}{14371412}a-\frac{1110313}{14371412}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{58257}{14371412}a^{7}-\frac{11037}{3592853}a^{6}-\frac{894903}{14371412}a^{5}-\frac{1021865}{7185706}a^{4}-\frac{4676523}{14371412}a^{3}+\frac{3856895}{14371412}a^{2}+\frac{23049217}{14371412}a+\frac{2220971}{14371412}$, $\frac{101825}{14371412}a^{7}-\frac{391139}{14371412}a^{6}-\frac{425778}{3592853}a^{5}+\frac{633510}{3592853}a^{4}+\frac{9535965}{14371412}a^{3}+\frac{3033009}{3592853}a^{2}+\frac{8428379}{14371412}a-\frac{7843643}{7185706}$, $\frac{5858}{326623}a^{7}-\frac{25649}{326623}a^{6}-\frac{25079}{326623}a^{5}-\frac{23791}{326623}a^{4}-\frac{170046}{326623}a^{3}+\frac{1137311}{326623}a^{2}-\frac{1063335}{326623}a-\frac{496378}{326623}$, $\frac{157885}{3592853}a^{7}-\frac{4260901}{14371412}a^{6}+\frac{2699889}{14371412}a^{5}+\frac{2128989}{7185706}a^{4}+\frac{4025979}{3592853}a^{3}+\frac{153167735}{14371412}a^{2}-\frac{110953387}{7185706}a-\frac{94144747}{14371412}$, $\frac{62795}{14371412}a^{7}+\frac{6605}{3592853}a^{6}-\frac{2159825}{14371412}a^{5}-\frac{590323}{7185706}a^{4}+\frac{1650163}{14371412}a^{3}+\frac{27519357}{14371412}a^{2}+\frac{114706631}{14371412}a+\frac{36729013}{14371412}$, $\frac{2960364457}{14371412}a^{7}-\frac{7366884507}{7185706}a^{6}-\frac{45315820593}{14371412}a^{5}+\frac{91981288301}{7185706}a^{4}+\frac{242826430289}{14371412}a^{3}-\frac{449628589763}{14371412}a^{2}-\frac{228401287211}{14371412}a-\frac{18405527715}{14371412}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 100453.738914 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 100453.738914 \cdot 1}{2\cdot\sqrt{321078613651227}}\cr\approx \mathstrut & 1.12717218453 \end{aligned}\]
Galois group
$S_4\wr C_2$ (as 8T47):
A solvable group of order 1152 |
The 20 conjugacy class representatives for $S_4\wr C_2$ |
Character table for $S_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{1857}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 36 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 | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.7.2 | $x^{8} + 6$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |
\(619\) | Deg $8$ | $2$ | $4$ | $4$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.619.2t1.a.a | $1$ | $ 619 $ | \(\Q(\sqrt{-619}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.1857.2t1.a.a | $1$ | $ 3 \cdot 619 $ | \(\Q(\sqrt{1857}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.5571.4t3.c.a | $2$ | $ 3^{2} \cdot 619 $ | 4.0.16713.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.16713.6t13.b.a | $4$ | $ 3^{3} \cdot 619 $ | 6.0.50139.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.10345347.12t34.b.a | $4$ | $ 3^{3} \cdot 619^{2}$ | 6.0.50139.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.6403769793.12t34.b.a | $4$ | $ 3^{3} \cdot 619^{3}$ | 6.0.50139.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.10345347.6t13.b.a | $4$ | $ 3^{3} \cdot 619^{2}$ | 6.0.50139.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.57633928137.12t201.a.a | $6$ | $ 3^{5} \cdot 619^{3}$ | 8.6.321078613651227.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
6.172901784411.12t202.a.a | $6$ | $ 3^{6} \cdot 619^{3}$ | 8.6.321078613651227.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-4$ | |
* | 6.172901784411.8t47.a.a | $6$ | $ 3^{6} \cdot 619^{3}$ | 8.6.321078613651227.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $4$ |
6.57633928137.12t200.a.a | $6$ | $ 3^{5} \cdot 619^{3}$ | 8.6.321078613651227.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
9.155...699.16t1294.a.a | $9$ | $ 3^{8} \cdot 619^{3}$ | 8.6.321078613651227.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $3$ | |
9.518705353233.18t272.a.a | $9$ | $ 3^{7} \cdot 619^{3}$ | 8.6.321078613651227.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-3$ | |
9.369...641.18t273.a.a | $9$ | $ 3^{8} \cdot 619^{6}$ | 8.6.321078613651227.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-3$ | |
9.123...547.18t274.a.a | $9$ | $ 3^{7} \cdot 619^{6}$ | 8.6.321078613651227.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $3$ | |
12.996...307.36t1763.a.a | $12$ | $ 3^{11} \cdot 619^{6}$ | 8.6.321078613651227.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.996...307.24t2821.a.a | $12$ | $ 3^{11} \cdot 619^{6}$ | 8.6.321078613651227.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.574...059.36t1758.a.a | $18$ | $ 3^{16} \cdot 619^{9}$ | 8.6.321078613651227.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |