Normalized defining polynomial
\( x^{9} - 18x^{7} - 36x^{6} + 144x^{4} + 420x^{3} + 216x^{2} - 432x - 296 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[5, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(74310349154112\) \(\medspace = 2^{6}\cdot 3^{22}\cdot 37\) | 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: | \(34.77\) | 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: | $2^{2/3}3^{22/9}37^{1/2}\approx 141.6067404717884$ | ||
Ramified primes: | \(2\), \(3\), \(37\) | 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: | \(\Q(\sqrt{37}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{4}a^{7}$, $\frac{1}{7278772}a^{8}+\frac{814663}{7278772}a^{7}-\frac{201929}{1819693}a^{6}-\frac{198350}{1819693}a^{5}+\frac{332350}{1819693}a^{4}+\frac{433539}{3639386}a^{3}-\frac{745441}{1819693}a^{2}-\frac{695825}{1819693}a+\frac{602505}{1819693}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $6$ | 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)
| |
Fundamental units: | $\frac{12015}{7278772}a^{8}+\frac{23649}{3639386}a^{7}-\frac{284971}{7278772}a^{6}-\frac{574533}{3639386}a^{5}-\frac{262077}{3639386}a^{4}+\frac{1009719}{3639386}a^{3}+\frac{1875024}{1819693}a^{2}+\frac{2971653}{1819693}a-\frac{1460872}{1819693}$, $\frac{15687}{7278772}a^{8}-\frac{42729}{3639386}a^{7}-\frac{118533}{7278772}a^{6}+\frac{158580}{1819693}a^{5}+\frac{154005}{1819693}a^{4}+\frac{366776}{1819693}a^{3}-\frac{385749}{1819693}a^{2}-\frac{2707854}{1819693}a+\frac{1830186}{1819693}$, $\frac{69771}{7278772}a^{8}-\frac{78375}{7278772}a^{7}-\frac{1080519}{7278772}a^{6}-\frac{312585}{1819693}a^{5}+\frac{43951}{1819693}a^{4}+\frac{1666108}{1819693}a^{3}+\frac{3940701}{1819693}a^{2}-\frac{2636221}{1819693}a-\frac{2991024}{1819693}$, $\frac{44220}{1819693}a^{8}-\frac{64461}{1819693}a^{7}-\frac{2888957}{7278772}a^{6}-\frac{466960}{1819693}a^{5}+\frac{885635}{1819693}a^{4}+\frac{4897036}{1819693}a^{3}+\frac{10449165}{1819693}a^{2}-\frac{6229331}{1819693}a-\frac{8334610}{1819693}$, $\frac{360061}{7278772}a^{8}-\frac{339039}{3639386}a^{7}-\frac{5515109}{7278772}a^{6}-\frac{608179}{1819693}a^{5}+\frac{4703647}{3639386}a^{4}+\frac{21777883}{3639386}a^{3}+\frac{18682529}{1819693}a^{2}-\frac{22310015}{1819693}a-\frac{21623892}{1819693}$, $\frac{13335}{7278772}a^{8}-\frac{36105}{7278772}a^{7}-\frac{129993}{7278772}a^{6}-\frac{146949}{3639386}a^{5}+\frac{49897}{3639386}a^{4}+\frac{1897597}{3639386}a^{3}+\frac{2346817}{1819693}a^{2}-\frac{211768}{1819693}a-\frac{3179806}{1819693}$ | 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: | \( 1539.48699795 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 1539.48699795 \cdot 3}{2\cdot\sqrt{74310349154112}}\cr\approx \mathstrut & 0.338417138134 \end{aligned}\]
Galois group
$S_3\wr C_3$ (as 9T28):
A solvable group of order 648 |
The 17 conjugacy class representatives for $S_3 \wr C_3 $ |
Character table for $S_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 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 siblings: | data not computed |
Degree 36 siblings: | 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 | R | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | R | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{5}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(3\) | 3.9.22.6 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
\(37\) | $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |