Normalized defining polynomial
\( x^{9} - 1413x^{7} + 665523x^{5} - 116096790x^{3} + 5468158809x - 4732879139 \)
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: | \(469965002851366869843441\) \(\medspace = 3^{22}\cdot 157^{6}\) | 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: | \(426.81\) | 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: | $3^{22/9}157^{2/3}\approx 426.80500345029895$ | ||
Ramified primes: | \(3\), \(157\) | 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: | \(4239=3^{3}\cdot 157\) | ||
Dirichlet character group: | $\lbrace$$\chi_{4239}(1,·)$, $\chi_{4239}(772,·)$, $\chi_{4239}(1414,·)$, $\chi_{4239}(3937,·)$, $\chi_{4239}(2185,·)$, $\chi_{4239}(2827,·)$, $\chi_{4239}(3598,·)$, $\chi_{4239}(1111,·)$, $\chi_{4239}(2524,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{157}a^{3}$, $\frac{1}{157}a^{4}$, $\frac{1}{338963}a^{5}+\frac{168}{338963}a^{4}-\frac{5}{2159}a^{3}-\frac{672}{2159}a^{2}+\frac{785}{2159}a+\frac{936}{2159}$, $\frac{1}{53217191}a^{6}-\frac{6}{338963}a^{4}+\frac{168}{338963}a^{3}+\frac{9}{2159}a^{2}-\frac{504}{2159}a-\frac{314}{2159}$, $\frac{1}{53217191}a^{7}-\frac{983}{338963}a^{4}+\frac{1021}{338963}a^{3}-\frac{218}{2159}a^{2}+\frac{78}{2159}a-\frac{861}{2159}$, $\frac{1}{53217191}a^{8}-\frac{78}{338963}a^{4}-\frac{574}{338963}a^{3}+\frac{156}{2159}a^{2}+\frac{31}{2159}a+\frac{354}{2159}$
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: | $\frac{1}{53217191}a^{6}-\frac{6}{338963}a^{4}+\frac{168}{338963}a^{3}+\frac{9}{2159}a^{2}-\frac{504}{2159}a+\frac{1845}{2159}$, $\frac{12}{53217191}a^{6}-\frac{72}{338963}a^{4}-\frac{143}{338963}a^{3}+\frac{108}{2159}a^{2}+\frac{429}{2159}a-\frac{1609}{2159}$, $\frac{3057}{53217191}a^{8}-\frac{72865}{53217191}a^{7}-\frac{2795503}{53217191}a^{6}+\frac{431808}{338963}a^{5}+\frac{4045412}{338963}a^{4}-\frac{101143822}{338963}a^{3}-\frac{1404548}{2159}a^{2}+\frac{35007115}{2159}a-\frac{29294404}{2159}$, $\frac{3562}{53217191}a^{8}+\frac{77308}{53217191}a^{7}-\frac{3429783}{53217191}a^{6}-\frac{472494}{338963}a^{5}+\frac{5256629}{338963}a^{4}+\frac{113612867}{338963}a^{3}-\frac{1668438}{2159}a^{2}-\frac{36453886}{2159}a+\frac{32897692}{2159}$, $\frac{5541}{53217191}a^{8}-\frac{82203}{53217191}a^{7}-\frac{6557887}{53217191}a^{6}+\frac{615108}{338963}a^{5}+\frac{13966144}{338963}a^{4}-\frac{201804213}{338963}a^{3}-\frac{6166385}{2159}a^{2}+\frac{80232445}{2159}a-\frac{25920926}{2159}$, $\frac{47632}{53217191}a^{8}+\frac{59514}{3130423}a^{7}-\frac{45798860}{53217191}a^{6}-\frac{6196100}{338963}a^{5}+\frac{70211337}{338963}a^{4}+\frac{1491245833}{338963}a^{3}-\frac{22455256}{2159}a^{2}-\frac{476845760}{2159}a+\frac{430262496}{2159}$, $\frac{1231}{53217191}a^{8}-\frac{33122}{53217191}a^{7}-\frac{903656}{53217191}a^{6}+\frac{166194}{338963}a^{5}+\frac{55447}{19939}a^{4}-\frac{33701779}{338963}a^{3}-\frac{173}{127}a^{2}+\frac{8905793}{2159}a-\frac{10744992}{2159}$, $\frac{669}{53217191}a^{8}-\frac{14159}{53217191}a^{7}-\frac{617131}{53217191}a^{6}+\frac{83187}{338963}a^{5}+\frac{905948}{338963}a^{4}-\frac{19241737}{338963}a^{3}-\frac{308621}{2159}a^{2}+\frac{6715174}{2159}a-\frac{5590185}{2159}$ (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: | \( 299546202.596 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 299546202.596 \cdot 3}{2\cdot\sqrt{469965002851366869843441}}\cr\approx \mathstrut & 0.335577071090 \end{aligned}\] (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 | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.1.0.1}{1} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
\(157\) | 157.9.6.2 | $x^{9} - 314 x^{6} + 24649 x^{3} + 89410007872$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ |