Normalized defining polynomial
\( x^{9} - 279x^{7} + 25947x^{5} - 893730x^{3} + 8311689x - 566029 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[9, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(27850805916667920729\) \(\medspace = 3^{22}\cdot 31^{6}\) | 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: | \(144.72\) | 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: | $3^{22/9}31^{2/3}\approx 144.7229910367496$ | ||
Ramified primes: | \(3\), \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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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: | \(837=3^{3}\cdot 31\) | ||
Dirichlet character group: | $\lbrace$$\chi_{837}(1,·)$, $\chi_{837}(67,·)$, $\chi_{837}(583,·)$, $\chi_{837}(559,·)$, $\chi_{837}(304,·)$, $\chi_{837}(625,·)$, $\chi_{837}(280,·)$, $\chi_{837}(25,·)$, $\chi_{837}(346,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{31}a^{3}$, $\frac{1}{31}a^{4}$, $\frac{1}{6169}a^{5}+\frac{35}{6169}a^{4}+\frac{44}{6169}a^{3}+\frac{59}{199}a^{2}-\frac{44}{199}a-\frac{19}{199}$, $\frac{1}{191239}a^{6}-\frac{6}{6169}a^{4}+\frac{35}{6169}a^{3}+\frac{9}{199}a^{2}+\frac{94}{199}a-\frac{62}{199}$, $\frac{1}{191239}a^{7}+\frac{46}{6169}a^{4}-\frac{54}{6169}a^{3}+\frac{50}{199}a^{2}+\frac{72}{199}a+\frac{85}{199}$, $\frac{1}{191239}a^{8}-\frac{72}{6169}a^{4}-\frac{76}{6169}a^{3}-\frac{55}{199}a^{2}-\frac{80}{199}a+\frac{78}{199}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
Rank: | $8$ | 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)
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Fundamental units: | $\frac{1}{191239}a^{6}-\frac{6}{6169}a^{4}+\frac{35}{6169}a^{3}+\frac{9}{199}a^{2}-\frac{105}{199}a-\frac{62}{199}$, $\frac{1}{191239}a^{6}-\frac{6}{6169}a^{4}+\frac{35}{6169}a^{3}+\frac{9}{199}a^{2}-\frac{105}{199}a+\frac{137}{199}$, $\frac{16}{191239}a^{8}+\frac{126}{191239}a^{7}-\frac{3590}{191239}a^{6}-\frac{903}{6169}a^{5}+\frac{7116}{6169}a^{4}+\frac{54851}{6169}a^{3}-\frac{2557}{199}a^{2}-\frac{19694}{199}a+\frac{1353}{199}$, $\frac{110}{191239}a^{7}+\frac{147}{191239}a^{6}-\frac{606}{6169}a^{5}-\frac{316}{6169}a^{4}+\frac{27067}{6169}a^{3}+\frac{123}{199}a^{2}-\frac{8910}{199}a+\frac{606}{199}$, $\frac{12}{191239}a^{8}+\frac{81}{191239}a^{7}-\frac{2324}{191239}a^{6}-\frac{470}{6169}a^{5}+\frac{3739}{6169}a^{4}+\frac{20253}{6169}a^{3}-\frac{1277}{199}a^{2}-\frac{6441}{199}a+\frac{445}{199}$, $\frac{26}{191239}a^{8}-\frac{114}{191239}a^{7}-\frac{5628}{191239}a^{6}+\frac{584}{6169}a^{5}+\frac{11670}{6169}a^{4}-\frac{14869}{6169}a^{3}-\frac{7605}{199}a^{2}-\frac{8811}{199}a+\frac{634}{199}$, $\frac{41}{191239}a^{8}+\frac{417}{191239}a^{7}-\frac{7336}{191239}a^{6}-\frac{2471}{6169}a^{5}+\frac{9780}{6169}a^{4}+\frac{110561}{6169}a^{3}-\frac{1780}{199}a^{2}-\frac{31144}{199}a+\frac{2129}{199}$, $\frac{26}{191239}a^{8}+\frac{196}{191239}a^{7}-\frac{5775}{191239}a^{6}-\frac{1391}{6169}a^{5}+\frac{11417}{6169}a^{4}+\frac{84699}{6169}a^{3}-\frac{4085}{199}a^{2}-\frac{30421}{199}a+\frac{1784}{199}$ (assuming GRH) | 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: | \( 2452297.55299 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 2452297.55299 \cdot 3}{2\cdot\sqrt{27850805916667920729}}\cr\approx \mathstrut & 0.356874487074 \end{aligned}\] (assuming GRH)
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 | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.1.0.1}{1} }^{9}$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | R | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 | $x^{9} + 9 x^{7} + 24 x^{6} + 18 x^{5} + 30$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
\(31\) | 31.9.6.3 | $x^{9} + 961 x^{3} - 834148$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ |