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Properties

Label 18.0.84451690958...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{31}\cdot 5^{9}\cdot 7^{14}\cdot 13^{14}$
Root discriminant $990.66$
Ramified primes $2, 3, 5, 7, 13$
Class number $73938211200$ (GRH)
Class group $[2, 2, 2, 2, 30, 154037940]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![499240711936, -788668438368, 736717649280, -491003942664, 257696335560, -107168086212, 33707047332, -7176130086, 966780294, -232194769, 149804439, -60135330, 13275402, -1530168, 53364, 6240, -552, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 552*x^16 + 6240*x^15 + 53364*x^14 - 1530168*x^13 + 13275402*x^12 - 60135330*x^11 + 149804439*x^10 - 232194769*x^9 + 966780294*x^8 - 7176130086*x^7 + 33707047332*x^6 - 107168086212*x^5 + 257696335560*x^4 - 491003942664*x^3 + 736717649280*x^2 - 788668438368*x + 499240711936)
 
gp: K = bnfinit(x^18 - 3*x^17 - 552*x^16 + 6240*x^15 + 53364*x^14 - 1530168*x^13 + 13275402*x^12 - 60135330*x^11 + 149804439*x^10 - 232194769*x^9 + 966780294*x^8 - 7176130086*x^7 + 33707047332*x^6 - 107168086212*x^5 + 257696335560*x^4 - 491003942664*x^3 + 736717649280*x^2 - 788668438368*x + 499240711936, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 552 x^{16} + 6240 x^{15} + 53364 x^{14} - 1530168 x^{13} + 13275402 x^{12} - 60135330 x^{11} + 149804439 x^{10} - 232194769 x^{9} + 966780294 x^{8} - 7176130086 x^{7} + 33707047332 x^{6} - 107168086212 x^{5} + 257696335560 x^{4} - 491003942664 x^{3} + 736717649280 x^{2} - 788668438368 x + 499240711936 \)

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:  \(-844516909583529949970724206594089321365947904000000000=-\,2^{18}\cdot 3^{31}\cdot 5^{9}\cdot 7^{14}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $990.66$
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:  $2, 3, 5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{72} a^{9} + \frac{1}{24} a^{8} + \frac{1}{24} a^{7} - \frac{1}{24} a^{6} + \frac{1}{12} a^{5} + \frac{1}{12} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{144} a^{10} - \frac{1}{24} a^{8} + \frac{1}{24} a^{7} - \frac{1}{48} a^{6} - \frac{5}{24} a^{5} + \frac{1}{12} a^{4} - \frac{1}{6} a^{3} - \frac{5}{12} a^{2} - \frac{1}{18} a - \frac{1}{3}$, $\frac{1}{288} a^{11} - \frac{1}{288} a^{10} - \frac{1}{144} a^{9} - \frac{1}{24} a^{8} + \frac{1}{96} a^{7} + \frac{11}{96} a^{6} + \frac{11}{48} a^{5} + \frac{1}{12} a^{4} + \frac{1}{24} a^{3} - \frac{29}{72} a^{2} + \frac{1}{36} a - \frac{4}{9}$, $\frac{1}{576} a^{12} - \frac{1}{576} a^{10} + \frac{1}{288} a^{9} - \frac{1}{64} a^{8} - \frac{1}{48} a^{7} + \frac{1}{64} a^{6} - \frac{7}{32} a^{5} - \frac{11}{48} a^{4} - \frac{13}{72} a^{3} - \frac{1}{16} a^{2} + \frac{25}{72} a - \frac{1}{9}$, $\frac{1}{1152} a^{13} - \frac{1}{1152} a^{11} + \frac{1}{576} a^{10} - \frac{1}{1152} a^{9} + \frac{1}{96} a^{8} + \frac{11}{384} a^{7} + \frac{23}{192} a^{6} - \frac{7}{96} a^{5} + \frac{29}{144} a^{4} - \frac{43}{96} a^{3} + \frac{1}{144} a^{2} + \frac{5}{18} a + \frac{4}{9}$, $\frac{1}{2304} a^{14} - \frac{1}{2304} a^{13} + \frac{1}{2304} a^{12} + \frac{1}{768} a^{11} + \frac{1}{768} a^{10} - \frac{5}{768} a^{9} - \frac{47}{768} a^{8} + \frac{11}{768} a^{7} - \frac{11}{192} a^{6} - \frac{23}{144} a^{5} - \frac{133}{576} a^{4} - \frac{65}{576} a^{3} - \frac{1}{48} a^{2} + \frac{1}{16} a + \frac{1}{6}$, $\frac{1}{2304} a^{15} - \frac{1}{1152} a^{11} + \frac{1}{192} a^{9} - \frac{1}{32} a^{8} + \frac{13}{256} a^{7} + \frac{5}{72} a^{6} + \frac{9}{64} a^{5} - \frac{5}{32} a^{4} + \frac{17}{192} a^{3} + \frac{13}{144} a^{2} + \frac{3}{16} a + \frac{1}{6}$, $\frac{1}{11722817862144} a^{16} - \frac{41772213}{1302535318016} a^{15} - \frac{2186122177}{11722817862144} a^{14} - \frac{1242443515}{11722817862144} a^{13} - \frac{1057820965}{3907605954048} a^{12} + \frac{5772253001}{3907605954048} a^{11} - \frac{27707057243}{11722817862144} a^{10} + \frac{16787252383}{11722817862144} a^{9} + \frac{32260680091}{651267659008} a^{8} + \frac{210876245381}{5861408931072} a^{7} + \frac{62099164909}{976901488512} a^{6} + \frac{431433938231}{2930704465536} a^{5} - \frac{204245178437}{1465352232768} a^{4} - \frac{99880063825}{488450744256} a^{3} - \frac{13789708439}{30528171516} a^{2} + \frac{45334654955}{366338058192} a - \frac{11335794119}{45792257274}$, $\frac{1}{26425397663510166232688226301685635209640197852147428738423808} a^{17} - \frac{517656700847028197117895970989062777819141944403}{13212698831755083116344113150842817604820098926073714369211904} a^{16} - \frac{904185164834395326036019277537303034322703258735588533439}{6606349415877541558172056575421408802410049463036857184605952} a^{15} + \frac{1200682132812357878950710476785902446634618046228957362973}{13212698831755083116344113150842817604820098926073714369211904} a^{14} + \frac{192191352546151922098551859364314129799609040439490202185}{825793676984692694771507071927676100301256182879607148075744} a^{13} - \frac{3852817872016799656226756305282229823565281931966848709649}{13212698831755083116344113150842817604820098926073714369211904} a^{12} + \frac{21572560306237833095880442732606939354478134632948785789015}{13212698831755083116344113150842817604820098926073714369211904} a^{11} + \frac{31569925288123787223901480018928865356163332615594579799595}{13212698831755083116344113150842817604820098926073714369211904} a^{10} + \frac{5022808121459587067424722757455467682085794184007508885491}{26425397663510166232688226301685635209640197852147428738423808} a^{9} + \frac{204884826842007610660687802876730829947024360262325792770133}{3303174707938770779086028287710704401205024731518428592302976} a^{8} + \frac{332724198943539011466731329590304313632654754219796785618875}{13212698831755083116344113150842817604820098926073714369211904} a^{7} + \frac{114343209862060050330775102797675335636891103451632950431429}{1651587353969385389543014143855352200602512365759214296151488} a^{6} - \frac{971243501456806545807055957221816403497289972623542493692041}{6606349415877541558172056575421408802410049463036857184605952} a^{5} + \frac{190783057921859795632752851966979288368490292329852437174069}{825793676984692694771507071927676100301256182879607148075744} a^{4} + \frac{583244193941037175169296767227940004581698925518712253954033}{3303174707938770779086028287710704401205024731518428592302976} a^{3} + \frac{160673915398600416224431221580038641154293198159989099345865}{825793676984692694771507071927676100301256182879607148075744} a^{2} - \frac{289188514261846667195626543492629505911241243179854347799249}{825793676984692694771507071927676100301256182879607148075744} a - \frac{400646302115231784560047593144375541369698179602419863117}{5432853138057188781391493894261026975666159097892152289972}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{30}\times C_{154037940}$, which has order $73938211200$ (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:  \( -1 \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1784332202668.8357 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.3.670761.2, 3.3.19656.1, 6.0.168720119670375.2, 6.0.144884376000.3, 9.9.379645737368741937160704.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$13$13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.12.10.3$x^{12} - 13 x^{6} + 2704$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$