Normalized defining polynomial
\( x^{9} - 171x^{7} + 9747x^{5} - 205770x^{3} + 1172889x - 733913 \)
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: | \(1476349596018920529\) \(\medspace = 3^{22}\cdot 19^{6}\) | 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: | \(104.42\) | 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: | $3^{22/9}19^{2/3}\approx 104.4236335758831$ | ||
Ramified primes: | \(3\), \(19\) | 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: | \(513=3^{3}\cdot 19\) | ||
Dirichlet character group: | $\lbrace$$\chi_{513}(1,·)$, $\chi_{513}(391,·)$, $\chi_{513}(7,·)$, $\chi_{513}(172,·)$, $\chi_{513}(49,·)$, $\chi_{513}(178,·)$, $\chi_{513}(343,·)$, $\chi_{513}(220,·)$, $\chi_{513}(349,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{19}a^{3}$, $\frac{1}{19}a^{4}$, $\frac{1}{1387}a^{5}+\frac{26}{1387}a^{4}-\frac{22}{1387}a^{3}-\frac{31}{73}a^{2}+\frac{22}{73}a-\frac{34}{73}$, $\frac{1}{26353}a^{6}-\frac{6}{1387}a^{4}+\frac{26}{1387}a^{3}+\frac{9}{73}a^{2}-\frac{5}{73}a+\frac{35}{73}$, $\frac{1}{26353}a^{7}+\frac{36}{1387}a^{4}-\frac{34}{1387}a^{3}+\frac{28}{73}a^{2}+\frac{21}{73}a+\frac{15}{73}$, $\frac{1}{26353}a^{8}-\frac{21}{1387}a^{4}+\frac{10}{1387}a^{3}-\frac{31}{73}a^{2}+\frac{26}{73}a-\frac{17}{73}$
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: | $\frac{2}{26353}a^{6}-\frac{12}{1387}a^{4}-\frac{21}{1387}a^{3}+\frac{18}{73}a^{2}+\frac{63}{73}a-\frac{76}{73}$, $\frac{2}{26353}a^{6}-\frac{12}{1387}a^{4}-\frac{21}{1387}a^{3}+\frac{18}{73}a^{2}+\frac{63}{73}a-\frac{3}{73}$, $\frac{7}{26353}a^{8}-\frac{53}{26353}a^{7}-\frac{784}{26353}a^{6}+\frac{309}{1387}a^{5}+\frac{1193}{1387}a^{4}-\frac{8739}{1387}a^{3}-\frac{451}{73}a^{2}+\frac{3217}{73}a-\frac{1849}{73}$, $\frac{3}{26353}a^{8}-\frac{28}{26353}a^{7}-\frac{271}{26353}a^{6}+\frac{149}{1387}a^{5}+\frac{195}{1387}a^{4}-\frac{3356}{1387}a^{3}+\frac{95}{73}a^{2}+\frac{619}{73}a-\frac{422}{73}$, $\frac{1}{26353}a^{8}+\frac{10}{26353}a^{7}-\frac{79}{26353}a^{6}-\frac{45}{1387}a^{5}+\frac{81}{1387}a^{4}+\frac{1015}{1387}a^{3}-\frac{16}{73}a^{2}-\frac{286}{73}a+\frac{66}{73}$, $\frac{1}{26353}a^{7}-\frac{2}{26353}a^{6}-\frac{5}{1387}a^{5}-\frac{9}{1387}a^{4}+\frac{170}{1387}a^{3}+\frac{19}{73}a^{2}-\frac{79}{73}a+\frac{42}{73}$, $\frac{4}{26353}a^{8}-\frac{29}{26353}a^{7}-\frac{401}{26353}a^{6}+\frac{149}{1387}a^{5}+\frac{553}{1387}a^{4}-\frac{3845}{1387}a^{3}-\frac{185}{73}a^{2}+\frac{1347}{73}a-\frac{770}{73}$, $\frac{1}{26353}a^{8}-\frac{6}{26353}a^{7}-\frac{128}{26353}a^{6}+\frac{29}{1387}a^{5}+\frac{263}{1387}a^{4}-\frac{467}{1387}a^{3}-\frac{206}{73}a^{2}-\frac{136}{73}a+\frac{194}{73}$ (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: | \( 399809.251389 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 399809.251389 \cdot 3}{2\cdot\sqrt{1476349596018920529}}\cr\approx \mathstrut & 0.252708269922 \end{aligned}\] (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 | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.22.2 | $x^{9} + 9 x^{7} + 24 x^{6} + 18 x^{5} + 30$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
\(19\) | 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |