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Properties

Label 18.0.99532033180...2711.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 7^{9}\cdot 11^{9}$
Root discriminant $31.61$
Ramified primes $3, 7, 11$
Class number $4$
Class group $[2, 2]$
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5929, -8085, 42, 12516, 6084, -25995, 32202, -30084, 20088, -8434, 1293, 870, -795, 378, -66, -36, 30, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 30*x^16 - 36*x^15 - 66*x^14 + 378*x^13 - 795*x^12 + 870*x^11 + 1293*x^10 - 8434*x^9 + 20088*x^8 - 30084*x^7 + 32202*x^6 - 25995*x^5 + 6084*x^4 + 12516*x^3 + 42*x^2 - 8085*x + 5929)
 
gp: K = bnfinit(x^18 - 9*x^17 + 30*x^16 - 36*x^15 - 66*x^14 + 378*x^13 - 795*x^12 + 870*x^11 + 1293*x^10 - 8434*x^9 + 20088*x^8 - 30084*x^7 + 32202*x^6 - 25995*x^5 + 6084*x^4 + 12516*x^3 + 42*x^2 - 8085*x + 5929, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 30 x^{16} - 36 x^{15} - 66 x^{14} + 378 x^{13} - 795 x^{12} + 870 x^{11} + 1293 x^{10} - 8434 x^{9} + 20088 x^{8} - 30084 x^{7} + 32202 x^{6} - 25995 x^{5} + 6084 x^{4} + 12516 x^{3} + 42 x^{2} - 8085 x + 5929 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-995320331802267761719062711=-\,3^{21}\cdot 7^{9}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.61$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{21} a^{8} + \frac{1}{7} a^{7} + \frac{2}{21} a^{6} + \frac{8}{21} a^{5} - \frac{3}{7} a^{4} + \frac{1}{3} a^{3} - \frac{5}{21} a^{2} - \frac{1}{3}$, $\frac{1}{63} a^{9} + \frac{1}{7} a^{6} - \frac{4}{21} a^{5} - \frac{5}{21} a^{4} + \frac{1}{7} a^{3} + \frac{5}{21} a^{2} + \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{63} a^{10} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{5}{21} a^{5} + \frac{1}{7} a^{4} - \frac{2}{21} a^{3} + \frac{1}{3} a^{2} + \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{63} a^{11} + \frac{1}{21} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{4} - \frac{1}{3} a^{3} + \frac{10}{63} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{2079} a^{12} - \frac{2}{693} a^{11} - \frac{4}{693} a^{10} + \frac{16}{2079} a^{9} - \frac{4}{693} a^{8} - \frac{5}{693} a^{7} - \frac{38}{231} a^{6} + \frac{163}{693} a^{5} + \frac{13}{63} a^{4} + \frac{604}{2079} a^{3} + \frac{19}{231} a^{2} + \frac{4}{11} a - \frac{4}{27}$, $\frac{1}{43659} a^{13} + \frac{4}{43659} a^{12} - \frac{79}{14553} a^{11} - \frac{137}{43659} a^{10} - \frac{116}{43659} a^{9} - \frac{16}{1617} a^{8} - \frac{1187}{14553} a^{7} + \frac{673}{14553} a^{6} - \frac{1114}{4851} a^{5} + \frac{12022}{43659} a^{4} + \frac{16507}{43659} a^{3} - \frac{310}{2079} a^{2} + \frac{2488}{6237} a - \frac{31}{81}$, $\frac{1}{43659} a^{14} - \frac{1}{43659} a^{12} - \frac{8}{43659} a^{11} + \frac{20}{4851} a^{10} - \frac{94}{43659} a^{9} - \frac{233}{14553} a^{8} - \frac{692}{4851} a^{7} - \frac{16}{2079} a^{6} - \frac{1361}{43659} a^{5} + \frac{38}{441} a^{4} - \frac{18043}{43659} a^{3} - \frac{244}{567} a^{2} + \frac{344}{693} a - \frac{20}{81}$, $\frac{1}{43659} a^{15} - \frac{4}{43659} a^{12} - \frac{19}{14553} a^{11} - \frac{1}{189} a^{10} - \frac{122}{43659} a^{9} - \frac{47}{4851} a^{8} + \frac{29}{4851} a^{7} + \frac{688}{6237} a^{6} - \frac{103}{539} a^{5} - \frac{177}{539} a^{4} + \frac{18509}{43659} a^{3} - \frac{268}{2079} a^{2} + \frac{316}{2079} a - \frac{22}{81}$, $\frac{1}{39424077} a^{16} - \frac{8}{39424077} a^{15} - \frac{4}{4380453} a^{14} + \frac{8}{804573} a^{13} - \frac{8422}{39424077} a^{12} + \frac{5008}{4380453} a^{11} + \frac{69514}{39424077} a^{10} - \frac{124655}{39424077} a^{9} + \frac{12658}{1460151} a^{8} + \frac{3613600}{39424077} a^{7} - \frac{2015252}{39424077} a^{6} - \frac{13399}{625779} a^{5} + \frac{7471580}{39424077} a^{4} + \frac{883727}{5632011} a^{3} - \frac{64913}{625779} a^{2} + \frac{140323}{804573} a - \frac{14152}{73143}$, $\frac{1}{1222146387} a^{17} + \frac{1}{174592341} a^{16} - \frac{955}{407382129} a^{15} + \frac{6173}{1222146387} a^{14} + \frac{11003}{1222146387} a^{13} - \frac{2507}{135794043} a^{12} - \frac{9244295}{1222146387} a^{11} + \frac{4580623}{1222146387} a^{10} - \frac{2789729}{407382129} a^{9} - \frac{22911866}{1222146387} a^{8} - \frac{11713853}{1222146387} a^{7} + \frac{59408141}{407382129} a^{6} - \frac{342888094}{1222146387} a^{5} - \frac{228871471}{1222146387} a^{4} + \frac{2946749}{19399149} a^{3} + \frac{4076813}{15872031} a^{2} - \frac{7971959}{24941763} a + \frac{62282}{755811}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3533553.0226 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-231}) \), 3.1.231.1 x3, 6.0.12326391.1, 9.1.2075751918009.1 x9

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 9 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R R ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$