Normalized defining polynomial
\( x^{8} - 2x^{7} - 11x^{6} + 5x^{5} + 33x^{4} + 52x^{3} - 141x^{2} + 35x + 25 \)
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: | \(-14294201135787\) \(\medspace = -\,3^{3}\cdot 853^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(44.10\) | 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: | $3^{3/4}853^{1/2}\approx 66.57565633599684$ | ||
Ramified primes: | \(3\), \(853\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{14635}a^{7}-\frac{6077}{14635}a^{6}-\frac{6341}{14635}a^{5}+\frac{452}{2927}a^{4}-\frac{1837}{14635}a^{3}-\frac{6678}{14635}a^{2}+\frac{489}{14635}a+\frac{53}{2927}$
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: | $\frac{2146}{14635}a^{7}-\frac{1457}{14635}a^{6}-\frac{26506}{14635}a^{5}-\frac{4699}{2927}a^{4}+\frac{53153}{14635}a^{3}+\frac{186932}{14635}a^{2}-\frac{92136}{14635}a-\frac{23831}{2927}$, $\frac{1903}{14635}a^{7}-\frac{2881}{14635}a^{6}-\frac{22318}{14635}a^{5}-\frac{382}{2927}a^{4}+\frac{60494}{14635}a^{3}+\frac{141296}{14635}a^{2}-\frac{181693}{14635}a-\frac{13294}{2927}$, $\frac{3367}{14635}a^{7}-\frac{1529}{14635}a^{6}-\frac{41587}{14635}a^{5}-\frac{8937}{2927}a^{4}+\frac{63966}{14635}a^{3}+\frac{287234}{14635}a^{2}-\frac{80467}{14635}a-\frac{26439}{2927}$, $\frac{5166}{14635}a^{7}-\frac{1707}{14635}a^{6}-\frac{63016}{14635}a^{5}-\frac{15349}{2927}a^{4}+\frac{81348}{14635}a^{3}+\frac{435197}{14635}a^{2}-\frac{64221}{14635}a-\frac{33537}{2927}$, $\frac{16094}{14635}a^{7}-\frac{12168}{14635}a^{6}-\frac{192454}{14635}a^{5}-\frac{31304}{2927}a^{4}+\frac{334627}{14635}a^{3}+\frac{1233048}{14635}a^{2}-\frac{706144}{14635}a-\frac{63169}{2927}$, $\frac{2101}{14635}a^{7}-\frac{6057}{14635}a^{6}-\frac{19226}{14635}a^{5}+\frac{7158}{2927}a^{4}+\frac{33373}{14635}a^{3}+\frac{33757}{14635}a^{2}-\frac{201951}{14635}a+\frac{23543}{2927}$ | 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: | \( 10179.92628 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 10179.92628 \cdot 1}{2\cdot\sqrt{14294201135787}}\cr\approx \mathstrut & 0.5413704357 \end{aligned}\]
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
\(\Q(\sqrt{853}) \) |
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 | ${\href{/padicField/2.8.0.1}{8} }$ | R | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(853\) | 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.2559.2t1.a.a | $1$ | $ 3 \cdot 853 $ | \(\Q(\sqrt{-2559}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.853.2t1.a.a | $1$ | $ 853 $ | \(\Q(\sqrt{853}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.2559.4t3.c.a | $2$ | $ 3 \cdot 853 $ | 4.0.7677.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.7677.6t13.a.a | $4$ | $ 3^{2} \cdot 853 $ | 6.0.23031.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.19645443.12t34.a.a | $4$ | $ 3^{3} \cdot 853^{2}$ | 6.0.23031.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.5585854293.12t34.a.a | $4$ | $ 3^{2} \cdot 853^{3}$ | 6.0.23031.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.2182827.6t13.a.a | $4$ | $ 3 \cdot 853^{2}$ | 6.0.23031.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.452454197733.12t201.a.a | $6$ | $ 3^{6} \cdot 853^{3}$ | 8.6.14294201135787.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
6.150818065911.12t202.a.a | $6$ | $ 3^{5} \cdot 853^{3}$ | 8.6.14294201135787.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-4$ | |
* | 6.16757562879.8t47.a.a | $6$ | $ 3^{3} \cdot 853^{3}$ | 8.6.14294201135787.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $4$ |
6.5585854293.12t200.a.a | $6$ | $ 3^{2} \cdot 853^{3}$ | 8.6.14294201135787.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
9.122...791.16t1294.a.a | $9$ | $ 3^{9} \cdot 853^{3}$ | 8.6.14294201135787.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $3$ | |
9.452454197733.18t272.a.a | $9$ | $ 3^{6} \cdot 853^{3}$ | 8.6.14294201135787.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-3$ | |
9.280...641.18t273.a.a | $9$ | $ 3^{6} \cdot 853^{6}$ | 8.6.14294201135787.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-3$ | |
9.758...307.18t274.a.a | $9$ | $ 3^{9} \cdot 853^{6}$ | 8.6.14294201135787.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $3$ | |
12.842...923.36t1763.a.a | $12$ | $ 3^{7} \cdot 853^{6}$ | 8.6.14294201135787.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.758...307.24t2821.a.a | $12$ | $ 3^{9} \cdot 853^{6}$ | 8.6.14294201135787.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.343...031.36t1758.a.a | $18$ | $ 3^{15} \cdot 853^{9}$ | 8.6.14294201135787.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |