Normalized defining polynomial
\( x^{9} - 15x^{7} - 4x^{6} + 54x^{5} + 12x^{4} - 38x^{3} - 9x^{2} + 6x + 1 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[9, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(62523502209\) \(\medspace = 3^{12}\cdot 7^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.83\) | 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: | $3^{4/3}7^{2/3}\approx 15.83289626371223$ | ||
Ramified primes: | \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $9$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(63=3^{2}\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{63}(1,·)$, $\chi_{63}(4,·)$, $\chi_{63}(37,·)$, $\chi_{63}(43,·)$, $\chi_{63}(46,·)$, $\chi_{63}(16,·)$, $\chi_{63}(22,·)$, $\chi_{63}(25,·)$, $\chi_{63}(58,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{508}a^{8}-\frac{69}{508}a^{7}+\frac{87}{254}a^{6}+\frac{91}{254}a^{5}+\frac{49}{127}a^{4}+\frac{51}{127}a^{3}+\frac{55}{254}a^{2}+\frac{21}{508}a+\frac{81}{508}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{369}{508}a^{8}-\frac{315}{508}a^{7}-\frac{2695}{254}a^{6}+\frac{1575}{254}a^{5}+\frac{4746}{127}a^{4}-\frac{2898}{127}a^{3}-\frac{4851}{254}a^{2}+\frac{4701}{508}a+\frac{1187}{508}$, $\frac{369}{508}a^{8}-\frac{315}{508}a^{7}-\frac{2695}{254}a^{6}+\frac{1575}{254}a^{5}+\frac{4746}{127}a^{4}-\frac{2898}{127}a^{3}-\frac{4851}{254}a^{2}+\frac{4701}{508}a+\frac{679}{508}$, $\frac{485}{254}a^{8}-\frac{159}{127}a^{7}-\frac{3525}{127}a^{6}+\frac{1336}{127}a^{5}+\frac{12097}{127}a^{4}-\frac{5013}{127}a^{3}-\frac{5456}{127}a^{2}+\frac{3073}{254}a+\frac{275}{127}$, $\frac{485}{254}a^{8}-\frac{159}{127}a^{7}-\frac{3525}{127}a^{6}+\frac{1336}{127}a^{5}+\frac{12097}{127}a^{4}-\frac{5013}{127}a^{3}-\frac{5456}{127}a^{2}+\frac{3073}{254}a+\frac{148}{127}$, $\frac{163}{254}a^{8}-\frac{99}{127}a^{7}-\frac{1186}{127}a^{6}+\frac{1117}{127}a^{5}+\frac{4290}{127}a^{4}-\frac{3948}{127}a^{3}-\frac{2592}{127}a^{2}+\frac{3423}{254}a+\frac{569}{127}$, $\frac{159}{127}a^{8}-\frac{225}{254}a^{7}-\frac{2306}{127}a^{6}+\frac{998}{127}a^{5}+\frac{7923}{127}a^{4}-\frac{3759}{127}a^{3}-\frac{3719}{127}a^{2}+\frac{1307}{127}a+\frac{231}{254}$, $\frac{1371}{508}a^{8}-\frac{873}{508}a^{7}-\frac{10009}{254}a^{6}+\frac{3603}{254}a^{5}+\frac{17395}{127}a^{4}-\frac{6787}{127}a^{3}-\frac{17559}{254}a^{2}+\frac{8471}{508}a+\frac{3101}{508}$, $\frac{1759}{508}a^{8}-\frac{975}{508}a^{7}-\frac{12829}{254}a^{6}+\frac{3605}{254}a^{5}+\frac{22056}{127}a^{4}-\frac{7192}{127}a^{3}-\frac{20857}{254}a^{2}+\frac{9507}{508}a+\frac{2271}{508}$ | 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: | \( 212.533159721 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 212.533159721 \cdot 1}{2\cdot\sqrt{62523502209}}\cr\approx \mathstrut & 0.217593048061 \end{aligned}\]
Galois group
An abelian group of order 9 |
The 9 conjugacy class representatives for $C_3^2$ |
Character table for $C_3^2$ |
Intermediate fields
\(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.12.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$ | $3$ | $3$ | $12$ | $C_3^2$ | $[2]^{3}$ |
\(7\) | 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.9.3t1.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) | $0$ | $1$ |
* | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.63.3t1.a.a | $1$ | $ 3^{2} \cdot 7 $ | 3.3.3969.2 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.63.3t1.b.a | $1$ | $ 3^{2} \cdot 7 $ | 3.3.3969.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.63.3t1.b.b | $1$ | $ 3^{2} \cdot 7 $ | 3.3.3969.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.63.3t1.a.b | $1$ | $ 3^{2} \cdot 7 $ | 3.3.3969.2 | $C_3$ (as 3T1) | $0$ | $1$ |