Normalized defining polynomial
\( x^{9} - 657 x^{7} - 657 x^{6} + 143883 x^{5} + 287766 x^{4} - 10423524 x^{3} - 31510377 x^{2} + \cdots + 6613289 \)
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: | \(4749028376057003861001\) \(\medspace = 3^{22}\cdot 73^{6}\) | 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: | \(256.16\) | 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: | $3^{22/9}73^{2/3}\approx 256.16019224397155$ | ||
Ramified primes: | \(3\), \(73\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{146}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{146}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{146}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{21316}a^{6}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{21316}a^{7}-\frac{1}{292}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{55997132}a^{8}-\frac{77}{27998566}a^{7}-\frac{2}{191771}a^{6}-\frac{351}{191771}a^{5}-\frac{2481}{767084}a^{4}+\frac{1237}{383542}a^{3}-\frac{2245}{10508}a^{2}-\frac{1080}{2627}a+\frac{1309}{5254}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{3}\times C_{3}$, which has order $9$ (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: | $\frac{1}{146}a^{3}-\frac{3}{2}a-\frac{5}{2}$, $\frac{1}{21316}a^{6}-\frac{3}{146}a^{4}-\frac{1}{73}a^{3}+\frac{9}{4}a^{2}+3a-\frac{1}{4}$, $\frac{1667}{27998566}a^{8}-\frac{48457}{55997132}a^{7}-\frac{360668}{13999283}a^{6}+\frac{64456}{191771}a^{5}+\frac{1286301}{383542}a^{4}-\frac{24775701}{767084}a^{3}-\frac{306829}{2627}a^{2}-\frac{718837}{10508}a+\frac{137363}{5254}$, $\frac{1543}{55997132}a^{8}+\frac{4956}{13999283}a^{7}-\frac{767135}{55997132}a^{6}-\frac{66537}{383542}a^{5}+\frac{1378531}{767084}a^{4}+\frac{8318571}{383542}a^{3}-\frac{61490}{2627}a^{2}-\frac{571903}{5254}a+\frac{301353}{10508}$, $\frac{10287}{55997132}a^{8}+\frac{36092}{13999283}a^{7}-\frac{4744021}{55997132}a^{6}-\frac{504235}{383542}a^{5}+\frac{6133303}{767084}a^{4}+\frac{63757121}{383542}a^{3}+\frac{2243279}{5254}a^{2}+\frac{1168261}{5254}a-\frac{906953}{10508}$, $\frac{591}{13999283}a^{8}+\frac{32621}{55997132}a^{7}-\frac{1004915}{55997132}a^{6}-\frac{52172}{191771}a^{5}+\frac{600773}{383542}a^{4}+\frac{24066781}{767084}a^{3}+\frac{769071}{10508}a^{2}+\frac{376957}{10508}a-\frac{149999}{10508}$, $\frac{5085}{13999283}a^{8}-\frac{124445}{55997132}a^{7}-\frac{2990656}{13999283}a^{6}+\frac{335321}{383542}a^{5}+\frac{16140853}{383542}a^{4}-\frac{55843791}{767084}a^{3}-\frac{14669561}{5254}a^{2}-\frac{27303315}{10508}a+\frac{4398013}{5254}$, $\frac{4}{191771}a^{8}-\frac{61657}{55997132}a^{7}-\frac{1788079}{55997132}a^{6}+\frac{81281}{191771}a^{5}+\frac{2149486}{191771}a^{4}-\frac{17797993}{767084}a^{3}-\frac{11840303}{10508}a^{2}-\frac{38495351}{10508}a-\frac{33113339}{10508}$ (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: | \( 14689322.6313 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 14689322.6313 \cdot 9}{2\cdot\sqrt{4749028376057003861001}}\cr\approx \mathstrut & 0.491113488332 \end{aligned}\] (assuming GRH)
Galois group
$C_3\wr C_3$ (as 9T17):
A solvable group of order 81 |
The 17 conjugacy class representatives for $C_3 \wr C_3 $ |
Character table for $C_3 \wr 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
Degree 9 siblings: | data not computed |
Degree 27 siblings: | data not computed |
Minimal sibling: | 9.9.167229666656361.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 | $x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$ | $9$ | $1$ | $22$ | $C_9:C_3$ | $[2, 3]^{3}$ |
\(73\) | 73.3.2.2 | $x^{3} + 292$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
73.3.2.3 | $x^{3} + 146$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |