Normalized defining polynomial
\( x^{9} + 3x^{7} - 3x^{6} + 12x^{4} - 36x^{3} + 36x^{2} - 24x + 8 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-7346640384\) \(\medspace = -\,2^{9}\cdot 3^{15}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(12.48\) | 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))
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Galois root discriminant: | $2^{3/2}3^{31/18}\approx 18.760906587882975$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-6}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}$, $\frac{1}{2308}a^{8}+\frac{49}{577}a^{7}+\frac{337}{2308}a^{6}+\frac{271}{2308}a^{5}-\frac{561}{1154}a^{4}-\frac{160}{577}a^{3}+\frac{155}{1154}a^{2}+\frac{197}{577}a-\frac{53}{577}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $\frac{9}{1154}a^{8}+\frac{33}{1154}a^{7}+\frac{74}{577}a^{6}+\frac{131}{1154}a^{5}+\frac{144}{577}a^{4}+\frac{5}{577}a^{3}-\frac{95}{1154}a^{2}+\frac{84}{577}a-\frac{954}{577}$, $\frac{39}{2308}a^{8}+\frac{143}{2308}a^{7}+\frac{449}{2308}a^{6}+\frac{190}{577}a^{5}+\frac{671}{2308}a^{4}+\frac{107}{577}a^{3}-\frac{151}{577}a^{2}+\frac{182}{577}a-\frac{336}{577}$, $\frac{122}{577}a^{8}+\frac{443}{2308}a^{7}+\frac{871}{1154}a^{6}+\frac{115}{2308}a^{5}+\frac{37}{2308}a^{4}+\frac{1546}{577}a^{3}-\frac{5717}{1154}a^{2}+\frac{1508}{577}a-\frac{476}{577}$, $\frac{169}{1154}a^{8}-\frac{107}{2308}a^{7}+\frac{407}{1154}a^{6}-\frac{1299}{2308}a^{5}-\frac{147}{2308}a^{4}+\frac{1312}{577}a^{3}-\frac{3232}{577}a^{2}+\frac{3693}{577}a-\frac{1181}{577}$, $\frac{169}{1154}a^{8}-\frac{107}{2308}a^{7}+\frac{407}{1154}a^{6}-\frac{1299}{2308}a^{5}-\frac{147}{2308}a^{4}+\frac{1312}{577}a^{3}-\frac{3232}{577}a^{2}+\frac{3116}{577}a-\frac{1181}{577}$ | 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: | \( 33.1180165264 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 33.1180165264 \cdot 1}{2\cdot\sqrt{7346640384}}\cr\approx \mathstrut & 0.383371235123 \end{aligned}\]
Galois group
$C_3\times S_3$ (as 9T4):
A solvable group of order 18 |
The 9 conjugacy class representatives for $S_3\times C_3$ |
Character table for $S_3\times C_3$ |
Intermediate fields
\(\Q(\zeta_{9})^+\), 3.1.216.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 6 sibling: | 6.0.10077696.3 |
Minimal sibling: | 6.0.10077696.3 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
\(3\) | 3.9.15.13 | $x^{9} + 3 x^{8} + 3 x^{7} + 3 x^{6} + 6 x^{3} + 3$ | $9$ | $1$ | $15$ | $S_3\times C_3$ | $[3/2, 2]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.24.2t1.b.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{-6}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
1.72.6t1.c.a | $1$ | $ 2^{3} \cdot 3^{2}$ | 6.0.10077696.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
* | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
1.72.6t1.c.b | $1$ | $ 2^{3} \cdot 3^{2}$ | 6.0.10077696.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
* | 2.216.3t2.b.a | $2$ | $ 2^{3} \cdot 3^{3}$ | 3.1.216.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.648.6t5.c.a | $2$ | $ 2^{3} \cdot 3^{4}$ | 9.3.7346640384.1 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |
* | 2.648.6t5.c.b | $2$ | $ 2^{3} \cdot 3^{4}$ | 9.3.7346640384.1 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |