Normalized defining polynomial
\( x^{9} - 4x^{7} - 3x^{6} - 18x^{5} + 8x^{4} - 24x^{3} + 18x^{2} - 4x - 1 \)
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: | \(-49633171875\) \(\medspace = -\,3^{3}\cdot 5^{6}\cdot 7^{6}\) | 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: | \(15.43\) | 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: | $3^{1/2}5^{2/3}7^{2/3}\approx 18.532726798013343$ | ||
Ramified primes: | \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{7}-\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{4}{9}a^{4}+\frac{2}{9}a^{3}-\frac{2}{9}a^{2}-\frac{4}{9}$, $\frac{1}{387}a^{8}+\frac{5}{129}a^{7}+\frac{49}{387}a^{6}+\frac{1}{387}a^{5}-\frac{89}{387}a^{4}+\frac{2}{129}a^{3}-\frac{20}{387}a^{2}+\frac{191}{387}a+\frac{152}{387}$
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: | $\frac{47}{387}a^{8}-\frac{26}{387}a^{7}-\frac{191}{387}a^{6}+\frac{4}{387}a^{5}-\frac{743}{387}a^{4}+\frac{626}{387}a^{3}-\frac{157}{43}a^{2}+\frac{1108}{387}a-\frac{170}{129}$, $\frac{47}{387}a^{8}-\frac{26}{387}a^{7}-\frac{191}{387}a^{6}+\frac{4}{387}a^{5}-\frac{743}{387}a^{4}+\frac{626}{387}a^{3}-\frac{157}{43}a^{2}+\frac{1108}{387}a-\frac{41}{129}$, $\frac{71}{387}a^{8}+\frac{119}{387}a^{7}-\frac{73}{129}a^{6}-\frac{220}{129}a^{5}-\frac{210}{43}a^{4}-\frac{1853}{387}a^{3}-\frac{1979}{387}a^{2}-\frac{1661}{387}a+\frac{386}{387}$, $\frac{4}{387}a^{8}+\frac{17}{387}a^{7}-\frac{19}{387}a^{6}-\frac{82}{387}a^{5}-\frac{55}{387}a^{4}-\frac{320}{387}a^{3}-\frac{41}{129}a^{2}-\frac{10}{387}a-\frac{41}{129}$, $\frac{97}{387}a^{8}+\frac{79}{387}a^{7}-\frac{107}{129}a^{6}-\frac{197}{129}a^{5}-\frac{742}{129}a^{4}-\frac{880}{387}a^{3}-\frac{3187}{387}a^{2}-\frac{49}{387}a-\frac{5}{387}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 53.7202008145 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 53.7202008145 \cdot 1}{2\cdot\sqrt{49633171875}}\cr\approx \mathstrut & 0.239249593597 \end{aligned}\]
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
\(\Q(\zeta_{7})^+\), 3.1.3675.1 |
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: | 6.0.826875.2 |
Minimal sibling: | 6.0.826875.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | R | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.1.0.1}{1} }^{9}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(5\) | 5.9.6.1 | $x^{9} + 9 x^{7} + 24 x^{6} + 27 x^{5} + 9 x^{4} - 186 x^{3} + 216 x^{2} - 504 x + 647$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(7\) | 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
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) | $1$ | $-1$ | |
* | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
1.21.6t1.a.a | $1$ | $ 3 \cdot 7 $ | 6.0.64827.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.21.6t1.a.b | $1$ | $ 3 \cdot 7 $ | 6.0.64827.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
* | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 2.3675.3t2.a.a | $2$ | $ 3 \cdot 5^{2} \cdot 7^{2}$ | 3.1.3675.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.525.6t5.a.a | $2$ | $ 3 \cdot 5^{2} \cdot 7 $ | 9.3.49633171875.1 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |
* | 2.525.6t5.a.b | $2$ | $ 3 \cdot 5^{2} \cdot 7 $ | 9.3.49633171875.1 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |