Normalized defining polynomial
\( x^{9} - 3x^{8} + 3x^{7} - 6x^{5} + 6x^{4} - 3x^{3} - 3x^{2} + 3x + 1 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(3831158169\) \(\medspace = 3^{16}\cdot 89\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.61\) | 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: | $3^{16/9}89^{1/2}\approx 66.51361729509448$ | ||
Ramified primes: | \(3\), \(89\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{89}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{17}a^{8}+\frac{3}{17}a^{6}-\frac{8}{17}a^{5}+\frac{4}{17}a^{4}+\frac{1}{17}a^{3}-\frac{3}{17}a-\frac{6}{17}$
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: | $a$, $\frac{15}{17}a^{8}-3a^{7}+\frac{62}{17}a^{6}-\frac{18}{17}a^{5}-\frac{76}{17}a^{4}+\frac{100}{17}a^{3}-4a^{2}-\frac{11}{17}a+\frac{29}{17}$, $\frac{3}{17}a^{8}-a^{7}+\frac{26}{17}a^{6}-\frac{7}{17}a^{5}-\frac{22}{17}a^{4}+\frac{54}{17}a^{3}-2a^{2}+\frac{8}{17}a+\frac{16}{17}$, $\frac{12}{17}a^{8}-2a^{7}+\frac{36}{17}a^{6}-\frac{11}{17}a^{5}-\frac{54}{17}a^{4}+\frac{63}{17}a^{3}-4a^{2}-\frac{2}{17}a+\frac{13}{17}$, $\frac{10}{17}a^{8}-2a^{7}+\frac{47}{17}a^{6}-\frac{29}{17}a^{5}-\frac{45}{17}a^{4}+\frac{78}{17}a^{3}-4a^{2}+\frac{4}{17}a+\frac{25}{17}$, $\frac{15}{17}a^{8}-3a^{7}+\frac{62}{17}a^{6}-\frac{18}{17}a^{5}-\frac{93}{17}a^{4}+\frac{134}{17}a^{3}-5a^{2}-\frac{11}{17}a+\frac{63}{17}$ | 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: | \( 27.0336794917 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 27.0336794917 \cdot 1}{2\cdot\sqrt{3831158169}}\cr\approx \mathstrut & 0.275879759573 \end{aligned}\]
Galois group
$S_3\wr C_3$ (as 9T28):
A solvable group of order 648 |
The 17 conjugacy class representatives for $S_3 \wr C_3 $ |
Character table for $S_3 \wr C_3 $ |
Intermediate fields
\(\Q(\zeta_{9})^+\) |
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 | ${\href{/padicField/2.9.0.1}{9} }$ | R | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.16.12 | $x^{9} + 6 x^{8} + 3$ | $9$ | $1$ | $16$ | $S_3\times C_3$ | $[2, 2]^{2}$ |
\(89\) | $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
89.3.0.1 | $x^{3} + 3 x + 86$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.89.2t1.a.a | $1$ | $ 89 $ | \(\Q(\sqrt{89}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.801.6t1.a.a | $1$ | $ 3^{2} \cdot 89 $ | 6.6.4625301609.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
* | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
1.801.6t1.a.b | $1$ | $ 3^{2} \cdot 89 $ | 6.6.4625301609.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
3.7209.6t6.a.a | $3$ | $ 3^{4} \cdot 89 $ | 6.2.583929.1 | $A_4\times C_2$ (as 6T6) | $1$ | $-1$ | |
3.641601.4t4.a.a | $3$ | $ 3^{4} \cdot 89^{2}$ | 4.0.641601.1 | $A_4$ (as 4T4) | $1$ | $-1$ | |
6.374649430329.18t197.a.a | $6$ | $ 3^{12} \cdot 89^{3}$ | 9.5.3831158169.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $-2$ | |
* | 6.47298249.9t28.a.a | $6$ | $ 3^{12} \cdot 89 $ | 9.5.3831158169.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $2$ |
6.296...009.18t202.a.a | $6$ | $ 3^{12} \cdot 89^{5}$ | 9.5.3831158169.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $2$ | |
6.374649430329.18t197.b.a | $6$ | $ 3^{12} \cdot 89^{3}$ | 9.5.3831158169.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $-2$ | |
8.333...281.12t176.a.a | $8$ | $ 3^{12} \cdot 89^{4}$ | 9.5.3831158169.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
8.270...761.24t1539.a.a | $8$ | $ 3^{16} \cdot 89^{4}$ | 9.5.3831158169.1 | $S_3 \wr C_3 $ (as 9T28) | $0$ | $0$ | |
8.270...761.24t1539.a.b | $8$ | $ 3^{16} \cdot 89^{4}$ | 9.5.3831158169.1 | $S_3 \wr C_3 $ (as 9T28) | $0$ | $0$ | |
12.218...641.18t206.a.a | $12$ | $ 3^{20} \cdot 89^{4}$ | 9.5.3831158169.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
12.137...481.36t1101.a.a | $12$ | $ 3^{20} \cdot 89^{8}$ | 9.5.3831158169.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ |