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Properties

Label 18.0.76944553869...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{33}\cdot 5^{16}\cdot 7^{12}$
Root discriminant $212.33$
Ramified primes $2, 3, 5, 7$
Class number $944784$ (GRH)
Class group $[3, 3, 3, 3, 9, 36, 36]$ (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![52743747, 0, 153013329, 0, 76305348, 0, 14934396, 0, 1031922, 0, -31914, 0, -5628, 0, -36, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 36*x^14 - 5628*x^12 - 31914*x^10 + 1031922*x^8 + 14934396*x^6 + 76305348*x^4 + 153013329*x^2 + 52743747)
 
gp: K = bnfinit(x^18 + 3*x^16 - 36*x^14 - 5628*x^12 - 31914*x^10 + 1031922*x^8 + 14934396*x^6 + 76305348*x^4 + 153013329*x^2 + 52743747, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} - 36 x^{14} - 5628 x^{12} - 31914 x^{10} + 1031922 x^{8} + 14934396 x^{6} + 76305348 x^{4} + 153013329 x^{2} + 52743747 \)

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:  \(-769445538694487691557611230000000000000000=-\,2^{16}\cdot 3^{33}\cdot 5^{16}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $212.33$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} + \frac{1}{16}$, $\frac{1}{32} a^{9} - \frac{1}{32} a^{8} + \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{5}{32} a + \frac{11}{32}$, $\frac{1}{64} a^{10} - \frac{1}{64} a^{8} - \frac{1}{16} a^{7} - \frac{1}{32} a^{6} - \frac{1}{16} a^{5} + \frac{1}{32} a^{4} + \frac{1}{16} a^{3} - \frac{15}{64} a^{2} - \frac{7}{16} a - \frac{17}{64}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{9} - \frac{1}{32} a^{7} - \frac{1}{16} a^{6} + \frac{1}{32} a^{5} - \frac{1}{16} a^{4} - \frac{15}{64} a^{3} + \frac{1}{16} a^{2} - \frac{17}{64} a - \frac{7}{16}$, $\frac{1}{128} a^{12} + \frac{1}{128} a^{8} - \frac{1}{16} a^{7} + \frac{1}{16} a^{5} - \frac{5}{128} a^{4} + \frac{1}{16} a^{3} + \frac{7}{16} a - \frac{61}{128}$, $\frac{1}{128} a^{13} + \frac{1}{128} a^{9} - \frac{1}{16} a^{6} - \frac{5}{128} a^{5} + \frac{1}{16} a^{4} + \frac{1}{16} a^{2} - \frac{61}{128} a + \frac{7}{16}$, $\frac{1}{256} a^{14} - \frac{1}{256} a^{12} + \frac{1}{256} a^{10} + \frac{7}{256} a^{8} - \frac{1}{16} a^{7} - \frac{5}{256} a^{6} - \frac{1}{16} a^{5} + \frac{21}{256} a^{4} + \frac{1}{16} a^{3} + \frac{3}{256} a^{2} + \frac{1}{16} a - \frac{27}{256}$, $\frac{1}{1792} a^{15} + \frac{3}{1792} a^{13} + \frac{13}{1792} a^{11} + \frac{1}{256} a^{9} - \frac{1}{32} a^{8} - \frac{29}{1792} a^{7} - \frac{151}{1792} a^{5} - \frac{1}{16} a^{4} + \frac{15}{1792} a^{3} - \frac{1}{4} a^{2} + \frac{75}{256} a - \frac{5}{32}$, $\frac{1}{207628214392738108928} a^{16} - \frac{67790214689831}{3244190849886532952} a^{14} + \frac{58804576667008549}{25953526799092263616} a^{12} + \frac{3095242854877111}{463455835698076136} a^{10} + \frac{2601014042864688321}{103814107196369054464} a^{8} - \frac{147413567975986327}{6488381699773065904} a^{6} - \frac{1}{8} a^{5} - \frac{373903941508269331}{6488381699773065904} a^{4} - \frac{1}{4} a^{3} + \frac{230874957853658813}{926911671396152272} a^{2} - \frac{1}{8} a - \frac{6923444637378328733}{29661173484676872704}$, $\frac{1}{124369300421250127247872} a^{17} + \frac{7124548830422883295}{31092325105312531811968} a^{15} + \frac{1796140676817116179}{4441760729330361687424} a^{13} + \frac{143264860533140464871}{31092325105312531811968} a^{11} + \frac{876099400374813685647}{62184650210625063623936} a^{9} - \frac{1}{32} a^{8} - \frac{905323712011292172435}{31092325105312531811968} a^{7} + \frac{762927468938466919609}{31092325105312531811968} a^{5} + \frac{1}{16} a^{4} - \frac{3580186164547917129363}{31092325105312531811968} a^{3} + \frac{3732006509856850898447}{17767042917321446749696} a + \frac{15}{32}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{36}\times C_{36}$, which has order $944784$ (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:  \( \frac{4021635}{15941978148418} a^{17} - \frac{218702233}{291510457571072} a^{15} - \frac{10758456105}{2040573202997504} a^{13} - \frac{2816977778715}{2040573202997504} a^{11} + \frac{337048028075}{2040573202997504} a^{9} + \frac{536926311526755}{2040573202997504} a^{7} + \frac{4415201073441189}{2040573202997504} a^{5} + \frac{11804647568975655}{2040573202997504} a^{3} + \frac{998866294608735}{291510457571072} a + \frac{1}{2} \) (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:  \( 252123542.90200993 \) (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

\(\Q(\sqrt{-3}) \), 3.1.2700.1 x3, 6.0.21870000.2, 9.1.506440367892900000000.2 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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
5Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$