Normalized defining polynomial
\( x^{9} - 504 x^{7} - 2646 x^{6} + 75411 x^{5} + 745731 x^{4} - 1569666 x^{3} - 46805094 x^{2} + \cdots - 305471341 \)
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: | \(481138652692603826361\) \(\medspace = 3^{22}\cdot 7^{6}\cdot 19^{4}\) | 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: | \(198.62\) | 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: | $3^{22/9}7^{2/3}19^{2/3}\approx 382.11799860557556$ | ||
Ramified primes: | \(3\), \(7\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{7}a^{3}$, $\frac{1}{7}a^{4}$, $\frac{1}{133}a^{5}+\frac{4}{133}a^{4}+\frac{6}{133}a^{3}$, $\frac{1}{931}a^{6}+\frac{4}{133}a^{4}+\frac{2}{133}a^{3}$, $\frac{1}{931}a^{7}+\frac{5}{133}a^{4}-\frac{5}{133}a^{3}$, $\frac{1}{185436504589}a^{8}-\frac{72104311}{185436504589}a^{7}-\frac{26778555}{185436504589}a^{6}+\frac{73903056}{26490929227}a^{5}-\frac{640304264}{26490929227}a^{4}-\frac{6060617}{199179919}a^{3}+\frac{53510500}{199179919}a^{2}-\frac{19636332}{199179919}a-\frac{90432992}{199179919}$
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: | $\frac{3092958}{185436504589}a^{8}-\frac{24815127}{185436504589}a^{7}-\frac{1354457353}{185436504589}a^{6}+\frac{372415167}{26490929227}a^{5}+\frac{4294770948}{3784418461}a^{4}+\frac{90650827402}{26490929227}a^{3}-\frac{10393470972}{199179919}a^{2}-\frac{72711559173}{199179919}a-\frac{131333084042}{199179919}$, $\frac{845721}{185436504589}a^{8}-\frac{2721867}{185436504589}a^{7}-\frac{429522855}{185436504589}a^{6}-\frac{97079310}{26490929227}a^{5}+\frac{10007883504}{26490929227}a^{4}+\frac{52601340923}{26490929227}a^{3}-\frac{3305164437}{199179919}a^{2}-\frac{30507936435}{199179919}a-\frac{63115163935}{199179919}$, $\frac{897397}{185436504589}a^{8}-\frac{6352370}{185436504589}a^{7}-\frac{48007689}{26490929227}a^{6}+\frac{5807937}{3784418461}a^{5}+\frac{5827642778}{26490929227}a^{4}+\frac{2710710144}{3784418461}a^{3}-\frac{1518762643}{199179919}a^{2}-\frac{10296169743}{199179919}a-\frac{17359817579}{199179919}$, $\frac{32025593}{185436504589}a^{8}-\frac{288161116}{185436504589}a^{7}-\frac{1918624334}{26490929227}a^{6}+\frac{4924306453}{26490929227}a^{5}+\frac{295420677198}{26490929227}a^{4}+\frac{113775377680}{3784418461}a^{3}-\frac{102863963337}{199179919}a^{2}-\frac{694137441623}{199179919}a-\frac{1231490641746}{199179919}$, $\frac{6852693}{9759816031}a^{8}-\frac{799551727}{185436504589}a^{7}-\frac{8656170799}{26490929227}a^{6}+\frac{592469540}{3784418461}a^{5}+\frac{72285108513}{1394259433}a^{4}+\frac{767489366552}{3784418461}a^{3}-\frac{467652071156}{199179919}a^{2}-\frac{3642703254035}{199179919}a-\frac{6907577591384}{199179919}$, $\frac{114007237}{26490929227}a^{8}-\frac{4926373432}{185436504589}a^{7}-\frac{53116876103}{26490929227}a^{6}+\frac{3742144960}{3784418461}a^{5}+\frac{8436282381818}{26490929227}a^{4}+\frac{247789162306}{199179919}a^{3}-\frac{2874957485852}{199179919}a^{2}-\frac{22382078500801}{199179919}a-\frac{42434834962710}{199179919}$, $\frac{108920803}{185436504589}a^{8}-\frac{1023929384}{185436504589}a^{7}-\frac{6484460753}{26490929227}a^{6}+\frac{19559468683}{26490929227}a^{5}+\frac{995562001388}{26490929227}a^{4}+\frac{17722925034}{199179919}a^{3}-\frac{349306842887}{199179919}a^{2}-\frac{2254442620293}{199179919}a-\frac{3909632011415}{199179919}$, $\frac{338121941}{26490929227}a^{8}-\frac{20167382320}{185436504589}a^{7}-\frac{145831673086}{26490929227}a^{6}+\frac{347475349946}{26490929227}a^{5}+\frac{22525826385480}{26490929227}a^{4}+\frac{453432732595}{199179919}a^{3}-\frac{7842123190630}{199179919}a^{2}-\frac{52194515087153}{199179919}a-\frac{91315352910940}{199179919}$ (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)]
| |
Regulator: | \( 9335475.77181 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 9335475.77181 \cdot 3}{2\cdot\sqrt{481138652692603826361}}\cr\approx \mathstrut & 0.326860584010 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 27 |
The 11 conjugacy class representatives for $C_9:C_3$ |
Character table for $C_9:C_3$ |
Intermediate fields
\(\Q(\zeta_{9})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.9.0.1}{9} }$ | R | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.22.11 | $x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$ | $9$ | $1$ | $22$ | $C_9:C_3$ | $[2, 3]^{3}$ |
\(7\) | 7.9.6.2 | $x^{9} + 252 x^{6} - 2352 x^{3} + 5488$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
19.3.2.2 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
19.3.2.2 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |