Normalized defining polynomial
\( x^{9} + 3x^{7} - 3x^{6} + 9x^{5} - 6x^{4} + 2x^{3} - 27x^{2} - 15x - 1 \)
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: | \(-45270270144\) \(\medspace = -\,2^{6}\cdot 3^{12}\cdot 11^{3}\) | 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: | \(15.27\) | 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: | $2^{2/3}3^{4/3}11^{1/2}\approx 22.779525808351707$ | ||
Ramified primes: | \(2\), \(3\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{71}a^{7}+\frac{10}{71}a^{6}+\frac{19}{71}a^{5}-\frac{14}{71}a^{4}-\frac{23}{71}a^{3}+\frac{17}{71}a^{2}-\frac{26}{71}a-\frac{11}{71}$, $\frac{1}{71}a^{8}-\frac{10}{71}a^{6}+\frac{9}{71}a^{5}-\frac{25}{71}a^{4}+\frac{34}{71}a^{3}+\frac{17}{71}a^{2}-\frac{35}{71}a-\frac{32}{71}$
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: | $\frac{14}{71}a^{8}-\frac{10}{71}a^{7}+\frac{44}{71}a^{6}-\frac{64}{71}a^{5}+\frac{145}{71}a^{4}-\frac{146}{71}a^{3}+\frac{68}{71}a^{2}-\frac{372}{71}a-\frac{54}{71}$, $\frac{9}{71}a^{8}-\frac{2}{71}a^{7}+\frac{32}{71}a^{6}-\frac{28}{71}a^{5}+\frac{87}{71}a^{4}-\frac{74}{71}a^{3}+\frac{48}{71}a^{2}-\frac{192}{71}a-\frac{53}{71}$, $\frac{22}{71}a^{8}+\frac{1}{71}a^{7}+\frac{74}{71}a^{6}-\frac{67}{71}a^{5}+\frac{217}{71}a^{4}-\frac{127}{71}a^{3}+\frac{107}{71}a^{2}-\frac{583}{71}a-\frac{360}{71}$, $\frac{12}{71}a^{8}-\frac{7}{71}a^{7}+\frac{23}{71}a^{6}-\frac{25}{71}a^{5}+\frac{82}{71}a^{4}-\frac{70}{71}a^{3}-\frac{57}{71}a^{2}-\frac{96}{71}a-\frac{23}{71}$, $\frac{1}{71}a^{8}-\frac{10}{71}a^{6}+\frac{9}{71}a^{5}-\frac{25}{71}a^{4}+\frac{34}{71}a^{3}-\frac{54}{71}a^{2}+\frac{36}{71}a+\frac{39}{71}$ | 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: | \( 61.5384431104 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 61.5384431104 \cdot 1}{2\cdot\sqrt{45270270144}}\cr\approx \mathstrut & 0.286972022146 \end{aligned}\]
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
\(\Q(\zeta_{9})^+\), 3.1.3564.1 |
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: | 6.0.1724976.1 |
Minimal sibling: | 6.0.1724976.1 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | R | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.1.0.1}{1} }^{9}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(3\) | 3.9.12.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$ | $3$ | $3$ | $12$ | $C_3^2$ | $[2]^{3}$ |
\(11\) | 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
1.99.6t1.a.a | $1$ | $ 3^{2} \cdot 11 $ | 6.0.8732691.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.99.6t1.a.b | $1$ | $ 3^{2} \cdot 11 $ | 6.0.8732691.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
* | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 2.3564.3t2.a.a | $2$ | $ 2^{2} \cdot 3^{4} \cdot 11 $ | 3.1.3564.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.396.6t5.c.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 11 $ | 9.3.45270270144.5 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |
* | 2.396.6t5.c.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 11 $ | 9.3.45270270144.5 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |