Normalized defining polynomial
\( x^{9} - 2x^{8} + 6x^{7} + 6x^{6} - 14x^{5} + 48x^{4} - 6x^{3} - 4x^{2} + 4x + 42 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(988800770304\) \(\medspace = 2^{8}\cdot 3^{3}\cdot 523^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.52\) | 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: | $2^{8/9}3^{1/2}523^{1/2}\approx 73.34891435042434$ | ||
Ramified primes: | \(2\), \(3\), \(523\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{1569}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{27}a^{7}+\frac{10}{27}a^{6}-\frac{2}{27}a^{5}-\frac{2}{27}a^{4}+\frac{2}{27}a^{3}+\frac{4}{27}a^{2}+\frac{2}{27}a-\frac{2}{9}$, $\frac{1}{21843}a^{8}-\frac{55}{7281}a^{7}+\frac{3304}{7281}a^{6}+\frac{1862}{7281}a^{5}-\frac{2834}{21843}a^{4}-\frac{6421}{21843}a^{3}+\frac{7861}{21843}a^{2}+\frac{4963}{21843}a+\frac{2978}{7281}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $4$ | 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: | $\frac{8}{21843}a^{8}-\frac{511}{21843}a^{7}+\frac{14}{21843}a^{6}-\frac{616}{21843}a^{5}-\frac{2447}{21843}a^{4}-\frac{6064}{21843}a^{3}+\frac{595}{21843}a^{2}-\frac{788}{7281}a+\frac{121}{2427}$, $\frac{109}{21843}a^{8}+\frac{622}{21843}a^{7}-\frac{416}{21843}a^{6}+\frac{3742}{21843}a^{5}+\frac{3368}{21843}a^{4}-\frac{7385}{21843}a^{3}+\frac{13871}{21843}a^{2}-\frac{3860}{7281}a+\frac{381}{809}$, $\frac{46}{7281}a^{8}-\frac{1736}{21843}a^{7}+\frac{5500}{21843}a^{6}-\frac{13862}{21843}a^{5}+\frac{3700}{21843}a^{4}+\frac{29690}{21843}a^{3}-\frac{119783}{21843}a^{2}+\frac{137201}{21843}a-\frac{31559}{7281}$, $\frac{1060}{21843}a^{8}-\frac{1774}{21843}a^{7}+\frac{5900}{21843}a^{6}+\frac{4943}{21843}a^{5}-\frac{2771}{7281}a^{4}+\frac{9121}{7281}a^{3}+\frac{445}{2427}a^{2}-\frac{28462}{21843}a-\frac{7319}{7281}$ | 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: | \( 435.153919807 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 435.153919807 \cdot 2}{2\cdot\sqrt{988800770304}}\cr\approx \mathstrut & 1.36407412529 \end{aligned}\]
Galois group
$S_3\wr S_3$ (as 9T31):
A solvable group of order 1296 |
The 22 conjugacy class representatives for $S_3\wr S_3$ |
Character table for $S_3\wr S_3$ |
Intermediate fields
3.1.2092.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 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 | R | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.9.8.1 | $x^{9} + 2$ | $9$ | $1$ | $8$ | $(C_9:C_3):C_2$ | $[\ ]_{9}^{6}$ |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(523\) | $\Q_{523}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{523}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{523}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.523.2t1.a.a | $1$ | $ 523 $ | \(\Q(\sqrt{-523}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.1569.2t1.a.a | $1$ | $ 3 \cdot 523 $ | \(\Q(\sqrt{1569}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.18828.6t3.a.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 523 $ | 6.2.61800048144.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.2092.3t2.a.a | $2$ | $ 2^{2} \cdot 523 $ | 3.1.2092.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.2092.4t5.a.a | $3$ | $ 2^{2} \cdot 523 $ | 4.2.2092.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.29541132.6t11.a.a | $3$ | $ 2^{2} \cdot 3^{3} \cdot 523^{2}$ | 6.0.118164528.3 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
3.1094116.6t8.a.a | $3$ | $ 2^{2} \cdot 523^{2}$ | 4.2.2092.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.56484.6t11.a.a | $3$ | $ 2^{2} \cdot 3^{3} \cdot 523 $ | 6.0.118164528.3 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
* | 6.472658112.9t31.a.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 523^{2}$ | 9.1.988800770304.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
6.129...248.18t300.a.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 523^{4}$ | 9.1.988800770304.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.129...248.18t319.a.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 523^{4}$ | 9.1.988800770304.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.472658112.18t311.a.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 523^{2}$ | 9.1.988800770304.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.424...304.24t2893.a.a | $8$ | $ 2^{8} \cdot 3^{4} \cdot 523^{6}$ | 9.1.988800770304.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.5671897344.12t213.a.a | $8$ | $ 2^{8} \cdot 3^{4} \cdot 523^{2}$ | 9.1.988800770304.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.798...712.36t2217.a.a | $12$ | $ 2^{10} \cdot 3^{6} \cdot 523^{7}$ | 9.1.988800770304.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
12.292...128.36t2214.a.a | $12$ | $ 2^{10} \cdot 3^{6} \cdot 523^{5}$ | 9.1.988800770304.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.611...776.36t2210.a.a | $12$ | $ 2^{12} \cdot 3^{6} \cdot 523^{6}$ | 9.1.988800770304.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.798...712.36t2216.a.a | $12$ | $ 2^{10} \cdot 3^{6} \cdot 523^{7}$ | 9.1.988800770304.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.292...128.18t315.a.a | $12$ | $ 2^{10} \cdot 3^{6} \cdot 523^{5}$ | 9.1.988800770304.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
16.601...144.24t2912.a.a | $16$ | $ 2^{14} \cdot 3^{8} \cdot 523^{8}$ | 9.1.988800770304.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |