Normalized defining polynomial
\( x^{9} - 1251x^{7} + 521667x^{5} - 80568570x^{3} + 3359709369x - 8172338617 \)
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: | \(226337443067263310114049\) \(\medspace = 3^{22}\cdot 139^{6}\) | 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: | \(393.53\) | 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: | $3^{22/9}139^{2/3}\approx 393.5255494187643$ | ||
Ramified primes: | \(3\), \(139\) | 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: | \(3753=3^{3}\cdot 139\) | ||
Dirichlet character group: | $\lbrace$$\chi_{3753}(1,·)$, $\chi_{3753}(1252,·)$, $\chi_{3753}(2503,·)$, $\chi_{3753}(1069,·)$, $\chi_{3753}(2320,·)$, $\chi_{3753}(3571,·)$, $\chi_{3753}(598,·)$, $\chi_{3753}(1849,·)$, $\chi_{3753}(3100,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{139}a^{3}$, $\frac{1}{139}a^{4}$, $\frac{1}{97717}a^{5}+\frac{204}{97717}a^{4}+\frac{8}{97717}a^{3}-\frac{113}{703}a^{2}-\frac{8}{703}a-\frac{231}{703}$, $\frac{1}{13582663}a^{6}-\frac{6}{97717}a^{4}+\frac{204}{97717}a^{3}+\frac{9}{703}a^{2}+\frac{91}{703}a-\frac{278}{703}$, $\frac{1}{13582663}a^{7}+\frac{22}{97717}a^{4}-\frac{107}{97717}a^{3}+\frac{116}{703}a^{2}-\frac{326}{703}a+\frac{20}{703}$, $\frac{1}{13582663}a^{8}+\frac{326}{97717}a^{4}-\frac{221}{97717}a^{3}+\frac{51}{703}a^{2}+\frac{196}{703}a+\frac{161}{703}$
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: | $\frac{3}{13582663}a^{6}-\frac{18}{97717}a^{4}-\frac{91}{97717}a^{3}+\frac{27}{703}a^{2}+\frac{273}{703}a-\frac{1537}{703}$, $\frac{10}{13582663}a^{6}-\frac{60}{97717}a^{4}-\frac{69}{97717}a^{3}+\frac{90}{703}a^{2}+\frac{207}{703}a-\frac{2780}{703}$, $\frac{22958}{13582663}a^{8}-\frac{194745}{13582663}a^{7}-\frac{26313196}{13582663}a^{6}+\frac{1573670}{97717}a^{5}+\frac{66496283}{97717}a^{4}-\frac{528597495}{97717}a^{3}-\frac{47044629}{703}a^{2}+\frac{316616730}{703}a-\frac{462086640}{703}$, $\frac{7339}{13582663}a^{8}+\frac{174446}{13582663}a^{7}-\frac{5045221}{13582663}a^{6}-\frac{865783}{97717}a^{5}+\frac{6967798}{97717}a^{4}+\frac{166876619}{97717}a^{3}-\frac{2000299}{703}a^{2}-\frac{48082056}{703}a+\frac{129901588}{703}$, $\frac{10185}{13582663}a^{8}-\frac{137984}{13582663}a^{7}-\frac{9357220}{13582663}a^{6}+\frac{36886}{5143}a^{5}+\frac{19946120}{97717}a^{4}-\frac{114794134}{97717}a^{3}-\frac{16448097}{703}a^{2}+\frac{13559907}{703}a+\frac{358943636}{703}$, $\frac{10353}{13582663}a^{8}+\frac{288434}{13582663}a^{7}-\frac{8760136}{13582663}a^{6}-\frac{1923293}{97717}a^{5}+\frac{12322305}{97717}a^{4}+\frac{467617489}{97717}a^{3}-\frac{1844272}{703}a^{2}-\frac{157879951}{703}a+\frac{403689316}{703}$, $\frac{109697}{13582663}a^{8}+\frac{2040511}{13582663}a^{7}-\frac{98977983}{13582663}a^{6}-\frac{696455}{5143}a^{5}+\frac{163459008}{97717}a^{4}+\frac{3031038717}{97717}a^{3}-\frac{47764846}{703}a^{2}-\frac{880403526}{703}a+\frac{2465680718}{703}$, $\frac{53816}{13582663}a^{8}-\frac{64150}{714877}a^{7}-\frac{39954058}{13582663}a^{6}+\frac{6542812}{97717}a^{5}+\frac{55151045}{97717}a^{4}-\frac{1273194447}{97717}a^{3}-\frac{19263747}{703}a^{2}+\frac{468752229}{703}a-\frac{1067869196}{703}$ (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: | \( 426399271.249 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 426399271.249 \cdot 3}{2\cdot\sqrt{226337443067263310114049}}\cr\approx \mathstrut & 0.688334395018 \end{aligned}\] (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 | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.1.0.1}{1} }^{9}$ | ${\href{/padicField/19.1.0.1}{1} }^{9}$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.1.0.1}{1} }^{9}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.22.6 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
\(139\) | 139.9.6.3 | $x^{9} + 115926 x^{3} - 367929803$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ |