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Properties

Label 18.0.29595566572...7968.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{53}\cdot 13^{12}$
Root discriminant $260.06$
Ramified primes $2, 3, 13$
Class number $81$ (GRH)
Class group $[3, 3, 3, 3]$ (GRH)
Galois group $D_9:C_3$ (as 18T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44484293569, -1898217, -7592787, 35433060, -26574498, -26575794, 35434770, -7595136, -1895301, 418685, 2907, -2304, 1554, -882, 414, -156, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 414*x^14 - 882*x^13 + 1554*x^12 - 2304*x^11 + 2907*x^10 + 418685*x^9 - 1895301*x^8 - 7595136*x^7 + 35434770*x^6 - 26575794*x^5 - 26574498*x^4 + 35433060*x^3 - 7592787*x^2 - 1898217*x + 44484293569)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 414*x^14 - 882*x^13 + 1554*x^12 - 2304*x^11 + 2907*x^10 + 418685*x^9 - 1895301*x^8 - 7595136*x^7 + 35434770*x^6 - 26575794*x^5 - 26574498*x^4 + 35433060*x^3 - 7592787*x^2 - 1898217*x + 44484293569, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 414 x^{14} - 882 x^{13} + 1554 x^{12} - 2304 x^{11} + 2907 x^{10} + 418685 x^{9} - 1895301 x^{8} - 7595136 x^{7} + 35434770 x^{6} - 26575794 x^{5} - 26574498 x^{4} + 35433060 x^{3} - 7592787 x^{2} - 1898217 x + 44484293569 \)

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:  \(-29595566572540406594298039314351025668947968=-\,2^{16}\cdot 3^{53}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $260.06$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{26} a^{6} - \frac{3}{26} a^{5} + \frac{3}{13} a^{4} - \frac{7}{26} a^{3} + \frac{3}{13} a^{2} - \frac{3}{26} a + \frac{1}{26}$, $\frac{1}{26} a^{7} - \frac{3}{26} a^{5} - \frac{1}{13} a^{4} + \frac{11}{26} a^{3} + \frac{1}{13} a^{2} - \frac{4}{13} a - \frac{5}{13}$, $\frac{1}{52} a^{8} + \frac{1}{26} a^{5} + \frac{3}{52} a^{4} - \frac{3}{26} a^{3} + \frac{5}{26} a^{2} - \frac{3}{26} a + \frac{3}{52}$, $\frac{1}{1404} a^{9} + \frac{1}{156} a^{8} - \frac{1}{78} a^{7} - \frac{2}{117} a^{6} + \frac{17}{156} a^{5} - \frac{37}{156} a^{4} + \frac{34}{117} a^{3} - \frac{19}{78} a^{2} + \frac{19}{156} a + \frac{259}{1404}$, $\frac{1}{1404} a^{10} + \frac{1}{156} a^{8} - \frac{2}{117} a^{7} - \frac{1}{156} a^{6} + \frac{7}{78} a^{5} - \frac{107}{468} a^{4} + \frac{23}{78} a^{3} - \frac{41}{156} a^{2} + \frac{251}{702} a - \frac{7}{156}$, $\frac{1}{1404} a^{11} + \frac{1}{468} a^{8} - \frac{1}{156} a^{7} + \frac{1}{78} a^{6} - \frac{2}{117} a^{5} + \frac{1}{156} a^{4} + \frac{1}{156} a^{3} + \frac{11}{54} a^{2} + \frac{1}{78} a - \frac{1}{156}$, $\frac{1}{36504} a^{12} - \frac{1}{6084} a^{11} - \frac{5}{36504} a^{10} + \frac{1}{18252} a^{9} + \frac{3}{338} a^{8} - \frac{73}{6084} a^{7} + \frac{59}{4056} a^{6} + \frac{19}{2028} a^{5} - \frac{19}{1521} a^{4} - \frac{3769}{18252} a^{3} + \frac{4141}{12168} a^{2} + \frac{1895}{18252} a + \frac{8399}{36504}$, $\frac{1}{36504} a^{13} + \frac{11}{36504} a^{11} - \frac{1}{18252} a^{10} - \frac{1}{18252} a^{9} + \frac{43}{6084} a^{8} + \frac{29}{12168} a^{7} - \frac{49}{6084} a^{6} + \frac{175}{6084} a^{5} - \frac{4045}{18252} a^{4} - \frac{275}{12168} a^{3} - \frac{7103}{18252} a^{2} + \frac{2773}{36504} a + \frac{8453}{18252}$, $\frac{1}{36504} a^{14} + \frac{1}{3042} a^{11} + \frac{1}{36504} a^{10} + \frac{1}{18252} a^{9} + \frac{37}{12168} a^{8} + \frac{2}{117} a^{7} + \frac{119}{12168} a^{6} + \frac{3707}{18252} a^{5} - \frac{3023}{12168} a^{4} + \frac{250}{1521} a^{3} - \frac{167}{1014} a^{2} + \frac{1817}{18252} a + \frac{4487}{36504}$, $\frac{1}{36504} a^{15} - \frac{5}{36504} a^{11} + \frac{5}{18252} a^{10} + \frac{1}{4056} a^{9} + \frac{41}{6084} a^{8} + \frac{181}{12168} a^{7} + \frac{287}{18252} a^{6} - \frac{2909}{12168} a^{5} + \frac{599}{6084} a^{4} + \frac{1555}{6084} a^{3} + \frac{5747}{18252} a^{2} + \frac{15037}{36504} a + \frac{2845}{6084}$, $\frac{1}{73008} a^{16} - \frac{5}{18252} a^{11} + \frac{5}{36504} a^{10} - \frac{1}{18252} a^{9} - \frac{215}{24336} a^{8} + \frac{259}{18252} a^{7} + \frac{223}{12168} a^{6} + \frac{25}{468} a^{5} - \frac{839}{4056} a^{4} - \frac{2755}{6084} a^{3} + \frac{4529}{36504} a^{2} - \frac{5165}{18252} a - \frac{13645}{73008}$, $\frac{1}{1175968602180014808149434131535442756192951952} a^{17} - \frac{6752911497561897385570863766940632161253}{1175968602180014808149434131535442756192951952} a^{16} - \frac{37148198113364160763151962193229497731}{146996075272501851018679266441930344524118994} a^{15} + \frac{1368240950455295439695177545079596488943}{146996075272501851018679266441930344524118994} a^{14} - \frac{506813004203537889402710514140383049293}{587984301090007404074717065767721378096475976} a^{13} + \frac{5117403389797207953560636892369052003621}{587984301090007404074717065767721378096475976} a^{12} + \frac{6887983444986419541068988868799087951758}{73498037636250925509339633220965172262059497} a^{11} + \frac{24328048593341188222994715762819784668233}{293992150545003702037358532883860689048237988} a^{10} + \frac{248871528628619490059165494208501452596895}{1175968602180014808149434131535442756192951952} a^{9} - \frac{3569041540761467718685025508483830747220551}{1175968602180014808149434131535442756192951952} a^{8} - \frac{40565163417460046561817833240554237318621}{73498037636250925509339633220965172262059497} a^{7} + \frac{1540373524477078134634846337781728019442309}{293992150545003702037358532883860689048237988} a^{6} + \frac{4643375678764886169076104922762518354419983}{587984301090007404074717065767721378096475976} a^{5} + \frac{64402550345130432503242878785345723740427891}{587984301090007404074717065767721378096475976} a^{4} - \frac{59291190172375722286112637305813933148466193}{146996075272501851018679266441930344524118994} a^{3} + \frac{118969564447210699605382623301595072173204429}{293992150545003702037358532883860689048237988} a^{2} - \frac{498565570272288973704610635244358575636097719}{1175968602180014808149434131535442756192951952} a + \frac{853981774531462747787565967531019156407}{5575609858946650079176883983137325609104}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}$, which has order $81$ (assuming GRH)

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:  \( \frac{804868271316470416903078653838}{16107393740138269890278245281824495345619} a^{17} - \frac{61348300938388460522600926961551}{144966543661244429012504207536420458110571} a^{16} + \frac{321284612450236329197001838095944}{144966543661244429012504207536420458110571} a^{15} - \frac{1182127686734374916713872910785232}{144966543661244429012504207536420458110571} a^{14} + \frac{3543416921817469437776640476344316}{144966543661244429012504207536420458110571} a^{13} - \frac{8813978229678501757695516846124588}{144966543661244429012504207536420458110571} a^{12} + \frac{19819092244648801244868481001006720}{144966543661244429012504207536420458110571} a^{11} + \frac{12207812913415099191356635891864424}{48322181220414809670834735845473486036857} a^{10} - \frac{101055934691223143923360121827054514}{48322181220414809670834735845473486036857} a^{9} + \frac{1607369572041900352475593014207766666}{48322181220414809670834735845473486036857} a^{8} - \frac{17192511906891348193703024435134255744}{144966543661244429012504207536420458110571} a^{7} - \frac{33509930699888253425845806198171559096}{144966543661244429012504207536420458110571} a^{6} + \frac{157483914260402549900227411894559752316}{144966543661244429012504207536420458110571} a^{5} - \frac{197885706366411089169546133272396238696}{144966543661244429012504207536420458110571} a^{4} + \frac{113289261487032376002306237195255921608}{144966543661244429012504207536420458110571} a^{3} - \frac{352582118474557033018375640685861592312}{144966543661244429012504207536420458110571} a^{2} + \frac{5472513952754822491558728261711430905898}{48322181220414809670834735845473486036857} a + \frac{32977727870597068648059066913759517}{76369847947439322802663872221363763} \) (order $6$)
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13089556510972.09 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9:C_3$ (as 18T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $D_9:C_3$
Character table for $D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.972.1 x3, 6.0.2834352.1, 9.1.3140889819384543273216.4 x3

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
2Data not computed
3Data not computed
$13$13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$