Normalized defining polynomial
\( x^{9} - 360x^{7} - 1557x^{6} + 34209x^{5} + 244350x^{4} - 549516x^{3} - 7960923x^{2} - 18280638x - 10581803 \)
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: | \(33680761597434559689\) \(\medspace = 3^{22}\cdot 181^{4}\) | 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: | \(147.81\) | 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: | $3^{22/9}181^{2/3}\approx 469.26206817515623$ | ||
Ramified primes: | \(3\), \(181\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{181}a^{6}+\frac{2}{181}a^{4}+\frac{72}{181}a^{3}$, $\frac{1}{181}a^{7}+\frac{2}{181}a^{5}+\frac{72}{181}a^{4}$, $\frac{1}{28\!\cdots\!51}a^{8}-\frac{1739598426292}{28\!\cdots\!51}a^{7}+\frac{1589867137749}{28\!\cdots\!51}a^{6}+\frac{11\!\cdots\!24}{28\!\cdots\!51}a^{5}-\frac{407899902447122}{28\!\cdots\!51}a^{4}+\frac{388636429448359}{28\!\cdots\!51}a^{3}-\frac{6314699663065}{15561019013471}a^{2}-\frac{4870281406989}{15561019013471}a-\frac{43117595635}{819001000709}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (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: | $\frac{33669515025}{28\!\cdots\!51}a^{8}-\frac{250515989127}{28\!\cdots\!51}a^{7}-\frac{10174924749070}{28\!\cdots\!51}a^{6}+\frac{22315607140023}{28\!\cdots\!51}a^{5}+\frac{963799952068635}{28\!\cdots\!51}a^{4}+\frac{12\!\cdots\!91}{28\!\cdots\!51}a^{3}-\frac{140202036703737}{15561019013471}a^{2}-\frac{459987713921625}{15561019013471}a-\frac{17059198472761}{819001000709}$, $\frac{7093169514}{28\!\cdots\!51}a^{8}-\frac{76742534241}{28\!\cdots\!51}a^{7}-\frac{1843519943012}{28\!\cdots\!51}a^{6}+\frac{10380686443722}{28\!\cdots\!51}a^{5}+\frac{159910366038276}{28\!\cdots\!51}a^{4}-\frac{213669438625665}{28\!\cdots\!51}a^{3}-\frac{23121514866114}{15561019013471}a^{2}-\frac{29510658150336}{15561019013471}a+\frac{1874667154404}{819001000709}$, $\frac{265759562088}{28\!\cdots\!51}a^{8}-\frac{2316848114875}{28\!\cdots\!51}a^{7}-\frac{75745773857808}{28\!\cdots\!51}a^{6}+\frac{243761212258368}{28\!\cdots\!51}a^{5}+\frac{70\!\cdots\!94}{28\!\cdots\!51}a^{4}+\frac{43\!\cdots\!10}{28\!\cdots\!51}a^{3}-\frac{10\!\cdots\!48}{15561019013471}a^{2}-\frac{28\!\cdots\!98}{15561019013471}a-\frac{94936246343531}{819001000709}$, $\frac{265759562088}{28\!\cdots\!51}a^{8}-\frac{2316848114875}{28\!\cdots\!51}a^{7}-\frac{75745773857808}{28\!\cdots\!51}a^{6}+\frac{243761212258368}{28\!\cdots\!51}a^{5}+\frac{70\!\cdots\!94}{28\!\cdots\!51}a^{4}+\frac{43\!\cdots\!10}{28\!\cdots\!51}a^{3}-\frac{10\!\cdots\!48}{15561019013471}a^{2}-\frac{28\!\cdots\!98}{15561019013471}a-\frac{94117245342822}{819001000709}$, $\frac{50696011141424}{28\!\cdots\!51}a^{8}-\frac{310893815772521}{28\!\cdots\!51}a^{7}-\frac{16\!\cdots\!38}{28\!\cdots\!51}a^{6}+\frac{21\!\cdots\!29}{28\!\cdots\!51}a^{5}+\frac{16\!\cdots\!33}{28\!\cdots\!51}a^{4}+\frac{25\!\cdots\!78}{28\!\cdots\!51}a^{3}-\frac{24\!\cdots\!93}{15561019013471}a^{2}-\frac{75\!\cdots\!11}{15561019013471}a-\frac{25\!\cdots\!52}{819001000709}$, $\frac{1253044617220}{28\!\cdots\!51}a^{8}+\frac{301983982246}{28\!\cdots\!51}a^{7}-\frac{375321699538576}{28\!\cdots\!51}a^{6}-\frac{21\!\cdots\!40}{28\!\cdots\!51}a^{5}+\frac{19\!\cdots\!69}{28\!\cdots\!51}a^{4}+\frac{22\!\cdots\!76}{28\!\cdots\!51}a^{3}+\frac{40\!\cdots\!14}{15561019013471}a^{2}+\frac{54\!\cdots\!82}{15561019013471}a+\frac{127096905894086}{819001000709}$, $\frac{14\!\cdots\!57}{28\!\cdots\!51}a^{8}-\frac{11\!\cdots\!19}{28\!\cdots\!51}a^{7}-\frac{43\!\cdots\!83}{28\!\cdots\!51}a^{6}+\frac{11\!\cdots\!55}{28\!\cdots\!51}a^{5}+\frac{40\!\cdots\!92}{28\!\cdots\!51}a^{4}+\frac{32\!\cdots\!89}{28\!\cdots\!51}a^{3}-\frac{58\!\cdots\!71}{15561019013471}a^{2}-\frac{17\!\cdots\!12}{15561019013471}a-\frac{55\!\cdots\!88}{819001000709}$, $\frac{10409211094183}{28\!\cdots\!51}a^{8}-\frac{1444821426480}{28\!\cdots\!51}a^{7}-\frac{27\!\cdots\!41}{28\!\cdots\!51}a^{6}-\frac{14\!\cdots\!72}{28\!\cdots\!51}a^{5}+\frac{10\!\cdots\!32}{28\!\cdots\!51}a^{4}+\frac{74\!\cdots\!95}{28\!\cdots\!51}a^{3}+\frac{17\!\cdots\!41}{15561019013471}a^{2}-\frac{15\!\cdots\!86}{15561019013471}a-\frac{698733290477625}{819001000709}$ (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: | \( 9972929.49103 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 9972929.49103 \cdot 1}{2\cdot\sqrt{33680761597434559689}}\cr\approx \mathstrut & 0.439918030374 \end{aligned}\] (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 | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.1.0.1}{1} }^{9}$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 | $x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$ | $9$ | $1$ | $22$ | $C_9:C_3$ | $[2, 3]^{3}$ |
\(181\) | $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
181.3.2.1 | $x^{3} + 181$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
181.3.2.1 | $x^{3} + 181$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |