Normalized defining polynomial
\( x^{8} - 12x^{6} - 8x^{5} + 5x^{4} + 48x^{3} + 202x^{2} + 124x + 8 \)
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: | \(-11917436282896\) \(\medspace = -\,2^{4}\cdot 929^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(43.10\) | 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: | $2^{4/3}929^{1/2}\approx 76.80353057714098$ | ||
Ramified primes: | \(2\), \(929\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{3748}a^{7}+\frac{269}{3748}a^{6}-\frac{737}{3748}a^{5}+\frac{383}{3748}a^{4}+\frac{459}{937}a^{3}-\frac{201}{937}a^{2}-\frac{141}{937}a-\frac{418}{937}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{7}{937}a^{7}+\frac{9}{937}a^{6}-\frac{11}{1874}a^{5}-\frac{130}{937}a^{4}-\frac{1469}{1874}a^{3}-\frac{6}{937}a^{2}+\frac{1674}{937}a+\frac{477}{937}$, $\frac{1751}{3748}a^{7}-\frac{73}{937}a^{6}-\frac{5213}{937}a^{5}-\frac{5283}{1874}a^{4}+\frac{9359}{3748}a^{3}+\frac{41013}{1874}a^{2}+\frac{170551}{1874}a+\frac{41107}{937}$, $\frac{1}{4}a^{7}-3a^{5}-2a^{4}+\frac{5}{4}a^{3}+12a^{2}+\frac{101}{2}a+30$, $\frac{1213}{3748}a^{7}-\frac{179}{937}a^{6}-\frac{7069}{1874}a^{5}-\frac{555}{1874}a^{4}+\frac{7315}{3748}a^{3}+\frac{26787}{1874}a^{2}+\frac{104883}{1874}a+\frac{3631}{937}$, $\frac{505}{3748}a^{7}-\frac{5}{937}a^{6}-\frac{2909}{1874}a^{5}-\frac{1073}{937}a^{4}+\frac{487}{3748}a^{3}+\frac{7187}{937}a^{2}+\frac{51549}{1874}a+\frac{13790}{937}$, $\frac{5323}{3748}a^{7}-\frac{13511}{1874}a^{6}-\frac{58947}{1874}a^{5}+\frac{17518}{937}a^{4}+\frac{460185}{3748}a^{3}+\frac{799523}{1874}a^{2}+\frac{1462645}{1874}a+\frac{543821}{937}$ | 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: | \( 54542.6592777 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 54542.6592777 \cdot 1}{2\cdot\sqrt{11917436282896}}\cr\approx \mathstrut & 3.17668926865 \end{aligned}\]
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
\(\Q(\sqrt{929}) \) |
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 | R | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(929\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $2$ | $3$ | $3$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3716.2t1.a.a | $1$ | $ 2^{2} \cdot 929 $ | \(\Q(\sqrt{-929}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.929.2t1.a.a | $1$ | $ 929 $ | \(\Q(\sqrt{929}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.3716.4t3.a.a | $2$ | $ 2^{2} \cdot 929 $ | 4.0.14864.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.59456.6t13.a.a | $4$ | $ 2^{6} \cdot 929 $ | 6.0.237824.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.55234624.12t34.a.a | $4$ | $ 2^{6} \cdot 929^{2}$ | 6.0.237824.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.51312965696.12t34.a.a | $4$ | $ 2^{6} \cdot 929^{3}$ | 6.0.237824.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.13808656.6t13.a.a | $4$ | $ 2^{4} \cdot 929^{2}$ | 6.0.237824.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.205251862784.12t201.a.a | $6$ | $ 2^{8} \cdot 929^{3}$ | 8.6.11917436282896.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
6.821007451136.12t202.a.a | $6$ | $ 2^{10} \cdot 929^{3}$ | 8.6.11917436282896.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-4$ | |
* | 6.12828241424.8t47.a.a | $6$ | $ 2^{4} \cdot 929^{3}$ | 8.6.11917436282896.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $4$ |
6.51312965696.12t200.a.a | $6$ | $ 2^{6} \cdot 929^{3}$ | 8.6.11917436282896.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
9.821007451136.16t1294.a.a | $9$ | $ 2^{10} \cdot 929^{3}$ | 8.6.11917436282896.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $3$ | |
9.131...176.18t272.a.a | $9$ | $ 2^{14} \cdot 929^{3}$ | 8.6.11917436282896.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-3$ | |
9.105...664.18t273.a.a | $9$ | $ 2^{14} \cdot 929^{6}$ | 8.6.11917436282896.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-3$ | |
9.658...104.18t274.a.a | $9$ | $ 2^{10} \cdot 929^{6}$ | 8.6.11917436282896.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $3$ | |
12.168...624.36t1763.a.a | $12$ | $ 2^{18} \cdot 929^{6}$ | 8.6.11917436282896.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.421...656.24t2821.a.a | $12$ | $ 2^{16} \cdot 929^{6}$ | 8.6.11917436282896.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.864...304.36t1758.a.a | $18$ | $ 2^{24} \cdot 929^{9}$ | 8.6.11917436282896.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |