Normalized defining polynomial
\( x^{9} - 117x^{7} + 4563x^{5} - 65910x^{3} + 257049x - 195533 \)
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: | \(151470380950257681\) \(\medspace = 3^{22}\cdot 13^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(81.08\) | 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: | $3^{22/9}13^{2/3}\approx 81.0821585694425$ | ||
Ramified primes: | \(3\), \(13\) | 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)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(351=3^{3}\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{351}(256,·)$, $\chi_{351}(1,·)$, $\chi_{351}(139,·)$, $\chi_{351}(133,·)$, $\chi_{351}(235,·)$, $\chi_{351}(16,·)$, $\chi_{351}(22,·)$, $\chi_{351}(250,·)$, $\chi_{351}(118,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{13}a^{3}$, $\frac{1}{13}a^{4}$, $\frac{1}{221}a^{5}-\frac{8}{221}a^{4}+\frac{3}{221}a^{3}-\frac{2}{17}a^{2}-\frac{3}{17}a-\frac{4}{17}$, $\frac{1}{2873}a^{6}-\frac{6}{221}a^{4}-\frac{8}{221}a^{3}-\frac{8}{17}a^{2}+\frac{7}{17}a+\frac{8}{17}$, $\frac{1}{2873}a^{7}-\frac{5}{221}a^{4}-\frac{1}{221}a^{3}-\frac{5}{17}a^{2}+\frac{7}{17}a-\frac{7}{17}$, $\frac{1}{2873}a^{8}-\frac{7}{221}a^{4}+\frac{1}{221}a^{3}-\frac{3}{17}a^{2}-\frac{5}{17}a-\frac{3}{17}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
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: | $\frac{1}{2873}a^{6}-\frac{6}{221}a^{4}-\frac{8}{221}a^{3}+\frac{9}{17}a^{2}+\frac{24}{17}a-\frac{43}{17}$, $\frac{1}{2873}a^{6}-\frac{6}{221}a^{4}-\frac{8}{221}a^{3}+\frac{9}{17}a^{2}+\frac{24}{17}a-\frac{26}{17}$, $\frac{1}{2873}a^{8}+\frac{20}{2873}a^{7}-\frac{108}{2873}a^{6}-\frac{126}{221}a^{5}+\frac{223}{221}a^{4}+\frac{2694}{221}a^{3}-\frac{58}{17}a^{2}-\frac{855}{17}a+\frac{704}{17}$, $\frac{1}{2873}a^{8}+\frac{7}{2873}a^{7}-\frac{75}{2873}a^{6}-\frac{47}{221}a^{5}+\frac{70}{221}a^{4}+\frac{1031}{221}a^{3}+\frac{95}{17}a^{2}-11a-\frac{158}{17}$, $\frac{1}{2873}a^{8}-\frac{6}{2873}a^{7}-\frac{79}{2873}a^{6}+\frac{35}{221}a^{5}+\frac{132}{221}a^{4}-\frac{701}{221}a^{3}-\frac{57}{17}a^{2}+\frac{264}{17}a-\frac{206}{17}$, $\frac{2}{2873}a^{8}-\frac{5}{2873}a^{7}-\frac{12}{169}a^{6}+\frac{42}{221}a^{5}+\frac{474}{221}a^{4}-\frac{1414}{221}a^{3}-\frac{269}{17}a^{2}+\frac{985}{17}a-\frac{700}{17}$, $\frac{2}{2873}a^{8}-\frac{7}{2873}a^{7}-\frac{212}{2873}a^{6}+\frac{54}{221}a^{5}+\frac{521}{221}a^{4}-\frac{1550}{221}a^{3}-\frac{338}{17}a^{2}+\frac{709}{17}a-\frac{288}{17}$, $\frac{4}{2873}a^{8}-\frac{18}{2873}a^{7}-\frac{380}{2873}a^{6}+\frac{125}{221}a^{5}+\frac{64}{17}a^{4}-\frac{3363}{221}a^{3}-\frac{481}{17}a^{2}+\frac{1919}{17}a-\frac{1250}{17}$ | 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: | \( 233016.463019 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 233016.463019 \cdot 3}{2\cdot\sqrt{151470380950257681}}\cr\approx \mathstrut & 0.459815881239 \end{aligned}\]
Galois group
A cyclic group of order 9 |
The 9 conjugacy class representatives for $C_9$ |
Character table for $C_9$ |
Intermediate fields
\(\Q(\zeta_{9})^+\) |
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 | ${\href{/padicField/2.9.0.1}{9} }$ | R | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | R | ${\href{/padicField/17.1.0.1}{1} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.1.0.1}{1} }^{9}$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.22.2 | $x^{9} + 9 x^{7} + 24 x^{6} + 18 x^{5} + 30$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
\(13\) | 13.9.6.3 | $x^{9} + 338 x^{3} - 24167$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.351.9t1.a.a | $1$ | $ 3^{3} \cdot 13 $ | 9.9.151470380950257681.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.351.9t1.a.b | $1$ | $ 3^{3} \cdot 13 $ | 9.9.151470380950257681.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.351.9t1.a.c | $1$ | $ 3^{3} \cdot 13 $ | 9.9.151470380950257681.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.351.9t1.a.d | $1$ | $ 3^{3} \cdot 13 $ | 9.9.151470380950257681.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.351.9t1.a.e | $1$ | $ 3^{3} \cdot 13 $ | 9.9.151470380950257681.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.351.9t1.a.f | $1$ | $ 3^{3} \cdot 13 $ | 9.9.151470380950257681.1 | $C_9$ (as 9T1) | $0$ | $1$ |