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Properties

Label 18.0.33689944996...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 5^{9}\cdot 13^{12}$
Root discriminant $106.98$
Ramified primes $2, 3, 5, 13$
Class number $134784$ (GRH)
Class group $[2, 2, 2, 36, 468]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![265747256, -91799112, 191003796, -51315152, 57659838, -12952074, 10180097, -2104020, 1196652, -243054, 104349, -22524, 7930, -1464, 402, -44, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 18*x^16 - 44*x^15 + 402*x^14 - 1464*x^13 + 7930*x^12 - 22524*x^11 + 104349*x^10 - 243054*x^9 + 1196652*x^8 - 2104020*x^7 + 10180097*x^6 - 12952074*x^5 + 57659838*x^4 - 51315152*x^3 + 191003796*x^2 - 91799112*x + 265747256)
 
gp: K = bnfinit(x^18 - 6*x^17 + 18*x^16 - 44*x^15 + 402*x^14 - 1464*x^13 + 7930*x^12 - 22524*x^11 + 104349*x^10 - 243054*x^9 + 1196652*x^8 - 2104020*x^7 + 10180097*x^6 - 12952074*x^5 + 57659838*x^4 - 51315152*x^3 + 191003796*x^2 - 91799112*x + 265747256, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 18 x^{16} - 44 x^{15} + 402 x^{14} - 1464 x^{13} + 7930 x^{12} - 22524 x^{11} + 104349 x^{10} - 243054 x^{9} + 1196652 x^{8} - 2104020 x^{7} + 10180097 x^{6} - 12952074 x^{5} + 57659838 x^{4} - 51315152 x^{3} + 191003796 x^{2} - 91799112 x + 265747256 \)

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:  \(-3368994499603041762677432832000000000=-\,2^{18}\cdot 3^{24}\cdot 5^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.98$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2340=2^{2}\cdot 3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2340}(1,·)$, $\chi_{2340}(841,·)$, $\chi_{2340}(139,·)$, $\chi_{2340}(781,·)$, $\chi_{2340}(79,·)$, $\chi_{2340}(1621,·)$, $\chi_{2340}(919,·)$, $\chi_{2340}(601,·)$, $\chi_{2340}(859,·)$, $\chi_{2340}(1699,·)$, $\chi_{2340}(1561,·)$, $\chi_{2340}(1381,·)$, $\chi_{2340}(679,·)$, $\chi_{2340}(1639,·)$, $\chi_{2340}(2161,·)$, $\chi_{2340}(1459,·)$, $\chi_{2340}(61,·)$, $\chi_{2340}(2239,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} + \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{11} + \frac{3}{8} a^{7} + \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} + \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{2967680392} a^{15} + \frac{87020491}{1483840196} a^{14} - \frac{143463739}{2967680392} a^{13} - \frac{111831945}{2967680392} a^{12} + \frac{86892797}{2967680392} a^{11} - \frac{62076069}{2967680392} a^{10} + \frac{55554895}{741920098} a^{9} - \frac{146744745}{2967680392} a^{8} - \frac{1272688349}{2967680392} a^{7} - \frac{360911719}{1483840196} a^{6} - \frac{133422171}{370960049} a^{5} + \frac{788828255}{2967680392} a^{4} + \frac{76776424}{370960049} a^{3} + \frac{577254763}{1483840196} a^{2} + \frac{233883269}{741920098} a + \frac{180916486}{370960049}$, $\frac{1}{5122216356592} a^{16} - \frac{1}{6811457921} a^{15} - \frac{6209605513}{640277044574} a^{14} + \frac{10079339635}{640277044574} a^{13} + \frac{40226629037}{2561108178296} a^{12} + \frac{19763582967}{640277044574} a^{11} + \frac{30317908513}{2561108178296} a^{10} - \frac{29953191947}{640277044574} a^{9} - \frac{513875740735}{5122216356592} a^{8} - \frac{145780869941}{320138522287} a^{7} - \frac{328780614001}{2561108178296} a^{6} + \frac{89716791738}{320138522287} a^{5} - \frac{1072085857303}{5122216356592} a^{4} + \frac{95396331029}{320138522287} a^{3} + \frac{60044801067}{1280554089148} a^{2} + \frac{191323352743}{640277044574} a - \frac{301018511709}{1280554089148}$, $\frac{1}{1176991062833215706321441821277327152} a^{17} - \frac{31107918899734171099}{73561941427075981645090113829832947} a^{16} + \frac{42809603893548134281844043}{294247765708303926580360455319331788} a^{15} + \frac{14527293390358109385822720403779417}{588495531416607853160720910638663576} a^{14} + \frac{9645962328134705142096712889321463}{294247765708303926580360455319331788} a^{13} - \frac{5320197251562044831007214727311081}{588495531416607853160720910638663576} a^{12} - \frac{8328357889285104189912375662392469}{147123882854151963290180227659665894} a^{11} - \frac{22816725139010778582849012303130355}{588495531416607853160720910638663576} a^{10} + \frac{38124093550129964542202976144332487}{1176991062833215706321441821277327152} a^{9} + \frac{14589260311965020828030360107271219}{147123882854151963290180227659665894} a^{8} + \frac{247089332413277005970326511678918913}{588495531416607853160720910638663576} a^{7} + \frac{176080340844213043150926239519783863}{588495531416607853160720910638663576} a^{6} + \frac{460931469293959988984579710608442631}{1176991062833215706321441821277327152} a^{5} + \frac{35600117516901253664012477300617027}{147123882854151963290180227659665894} a^{4} + \frac{20553628024077205828211993203927951}{73561941427075981645090113829832947} a^{3} + \frac{96280180751985255437228348389204337}{294247765708303926580360455319331788} a^{2} + \frac{34556034332882102722327139321021013}{294247765708303926580360455319331788} a + \frac{6752477403469664167174349166320349}{147123882854151963290180227659665894}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{36}\times C_{468}$, which has order $134784$ (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:  \( 400417.1364448253 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\zeta_{9})^+\), 3.3.13689.2, 3.3.169.1, 3.3.13689.1, 6.0.52488000.1, 6.0.1499109768000.6, 6.0.228488000.1, 6.0.1499109768000.4, 9.9.2565164201769.1

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

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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13Data not computed