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Properties

Label 18.0.85191817418...2544.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{9}\cdot 7^{12}\cdot 13^{12}$
Root discriminant $99.11$
Ramified primes $2, 3, 7, 13$
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![652602712, 95100288, 356716588, 51485984, 93696170, 12244424, 15348721, 1558324, 1740052, 123252, 146137, 6384, 9178, 124, 462, -16, 14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 14*x^16 - 16*x^15 + 462*x^14 + 124*x^13 + 9178*x^12 + 6384*x^11 + 146137*x^10 + 123252*x^9 + 1740052*x^8 + 1558324*x^7 + 15348721*x^6 + 12244424*x^5 + 93696170*x^4 + 51485984*x^3 + 356716588*x^2 + 95100288*x + 652602712)
 
gp: K = bnfinit(x^18 + 14*x^16 - 16*x^15 + 462*x^14 + 124*x^13 + 9178*x^12 + 6384*x^11 + 146137*x^10 + 123252*x^9 + 1740052*x^8 + 1558324*x^7 + 15348721*x^6 + 12244424*x^5 + 93696170*x^4 + 51485984*x^3 + 356716588*x^2 + 95100288*x + 652602712, 1)
 

Normalized defining polynomial

\( x^{18} + 14 x^{16} - 16 x^{15} + 462 x^{14} + 124 x^{13} + 9178 x^{12} + 6384 x^{11} + 146137 x^{10} + 123252 x^{9} + 1740052 x^{8} + 1558324 x^{7} + 15348721 x^{6} + 12244424 x^{5} + 93696170 x^{4} + 51485984 x^{3} + 356716588 x^{2} + 95100288 x + 652602712 \)

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:  \(-851918174186670722606904836351852544=-\,2^{27}\cdot 3^{9}\cdot 7^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.11$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2184=2^{3}\cdot 3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2184}(1,·)$, $\chi_{2184}(1537,·)$, $\chi_{2184}(841,·)$, $\chi_{2184}(653,·)$, $\chi_{2184}(1901,·)$, $\chi_{2184}(529,·)$, $\chi_{2184}(1849,·)$, $\chi_{2184}(989,·)$, $\chi_{2184}(289,·)$, $\chi_{2184}(1829,·)$, $\chi_{2184}(1873,·)$, $\chi_{2184}(365,·)$, $\chi_{2184}(29,·)$, $\chi_{2184}(625,·)$, $\chi_{2184}(1205,·)$, $\chi_{2184}(1465,·)$, $\chi_{2184}(893,·)$, $\chi_{2184}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{10} + \frac{1}{20} a^{9} - \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{10} a^{6} - \frac{1}{4} a^{5} + \frac{2}{5} a^{4} - \frac{7}{20} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{10}$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{11} + \frac{1}{20} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{4} a^{6} + \frac{2}{5} a^{5} - \frac{7}{20} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{10} a$, $\frac{1}{60} a^{14} - \frac{1}{60} a^{13} + \frac{7}{60} a^{11} - \frac{1}{15} a^{10} + \frac{1}{10} a^{9} + \frac{1}{20} a^{8} - \frac{1}{2} a^{7} - \frac{19}{60} a^{6} + \frac{1}{6} a^{5} - \frac{1}{4} a^{4} - \frac{7}{60} a^{3} - \frac{7}{15} a^{2} - \frac{11}{30} a - \frac{2}{15}$, $\frac{1}{5495809140} a^{15} - \frac{11874767}{1831936380} a^{14} - \frac{6795187}{5495809140} a^{13} - \frac{33447989}{1373952285} a^{12} + \frac{30725141}{1831936380} a^{11} + \frac{199121611}{2747904570} a^{10} + \frac{7850672}{457984095} a^{9} - \frac{15154091}{122129092} a^{8} + \frac{1907269301}{5495809140} a^{7} - \frac{323433073}{1831936380} a^{6} - \frac{686297159}{1373952285} a^{5} - \frac{1116028387}{5495809140} a^{4} + \frac{251690279}{2747904570} a^{3} + \frac{412626562}{1373952285} a^{2} - \frac{52927571}{457984095} a - \frac{235847624}{1373952285}$, $\frac{1}{11936897452080} a^{16} - \frac{83}{2984224363020} a^{15} + \frac{11639721139}{1492112181510} a^{14} - \frac{762688151}{55263414130} a^{13} - \frac{55392056917}{5968448726040} a^{12} + \frac{5886620747}{51452144190} a^{11} + \frac{392318503957}{5968448726040} a^{10} + \frac{27203434411}{497370727170} a^{9} + \frac{592706603729}{11936897452080} a^{8} - \frac{469768569439}{2984224363020} a^{7} - \frac{533658791599}{5968448726040} a^{6} + \frac{40683957899}{248685363585} a^{5} - \frac{97194415577}{3978965817360} a^{4} - \frac{678284639}{596844872604} a^{3} - \frac{703642156213}{1492112181510} a^{2} - \frac{336475012081}{1492112181510} a + \frac{1121295906799}{2984224363020}$, $\frac{1}{655965996005649624453912253875549840} a^{17} - \frac{349896441926246800549}{40997874750353101528369515867221865} a^{16} - \frac{656982699905464240255543}{13665958250117700509456505289073955} a^{15} - \frac{1174710265949964322225667960610769}{163991499001412406113478063468887460} a^{14} + \frac{2354655887914433166869174545372343}{327982998002824812226956126937774920} a^{13} - \frac{65696711522651787262982332965503}{9110638833411800339637670192715970} a^{12} + \frac{32721197144402755006592559647706881}{327982998002824812226956126937774920} a^{11} + \frac{352262806326207129783777505600180}{8199574950070620305673903173444373} a^{10} + \frac{16925336225814151697119399440436373}{655965996005649624453912253875549840} a^{9} + \frac{2121012149320810128960791249371001}{32798299800282481222695612693777492} a^{8} + \frac{1605305466726305931912971321616799}{3769919517273848416401794562503160} a^{7} + \frac{50875558436873105767872120250922197}{163991499001412406113478063468887460} a^{6} - \frac{70481264573968313264055191820699013}{218655332001883208151304084625183280} a^{5} - \frac{15471322795070548224940044874155589}{163991499001412406113478063468887460} a^{4} + \frac{24666716667973217611553958674189867}{54663833000470802037826021156295820} a^{3} + \frac{20076246793171459484620578984752611}{81995749500706203056739031734443730} a^{2} + \frac{33353892931991470498831713481003937}{163991499001412406113478063468887460} a - \frac{3987426969548855837049847590892642}{8199574950070620305673903173444373}$

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:  \( 205236.825908 \) (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

\(\Q(\sqrt{-6}) \), 3.3.8281.1, 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, 6.0.947980260864.5, 6.0.394827264.1, 6.0.33191424.1, 6.0.947980260864.4, 9.9.567869252041.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$