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Properties

Label 18.0.45028390589...0000.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{37}\cdot 5^{12}$
Root discriminant $44.40$
Ramified primes $2, 3, 5$
Class number $27$ (GRH)
Class group $[3, 3, 3]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4225, -2925, -9675, 32460, -1530, -41490, 45834, -30852, 17235, -7439, 1791, 36, -42, -198, 234, -132, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 132*x^15 + 234*x^14 - 198*x^13 - 42*x^12 + 36*x^11 + 1791*x^10 - 7439*x^9 + 17235*x^8 - 30852*x^7 + 45834*x^6 - 41490*x^5 - 1530*x^4 + 32460*x^3 - 9675*x^2 - 2925*x + 4225)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 132*x^15 + 234*x^14 - 198*x^13 - 42*x^12 + 36*x^11 + 1791*x^10 - 7439*x^9 + 17235*x^8 - 30852*x^7 + 45834*x^6 - 41490*x^5 - 1530*x^4 + 32460*x^3 - 9675*x^2 - 2925*x + 4225, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 132 x^{15} + 234 x^{14} - 198 x^{13} - 42 x^{12} + 36 x^{11} + 1791 x^{10} - 7439 x^{9} + 17235 x^{8} - 30852 x^{7} + 45834 x^{6} - 41490 x^{5} - 1530 x^{4} + 32460 x^{3} - 9675 x^{2} - 2925 x + 4225 \)

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:  \(-450283905890997363000000000000=-\,2^{12}\cdot 3^{37}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.40$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{36} a^{9} + \frac{1}{6} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{4} a - \frac{4}{9}$, $\frac{1}{72} a^{10} - \frac{1}{72} a^{9} - \frac{1}{8} a^{8} + \frac{1}{12} a^{7} + \frac{1}{24} a^{6} - \frac{1}{8} a^{5} + \frac{5}{24} a^{4} + \frac{1}{6} a^{3} - \frac{3}{8} a^{2} + \frac{29}{72} a - \frac{11}{72}$, $\frac{1}{72} a^{11} - \frac{1}{24} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{3} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{36} a^{2} - \frac{3}{8}$, $\frac{1}{1872} a^{12} - \frac{1}{156} a^{11} + \frac{1}{1872} a^{10} + \frac{7}{936} a^{9} - \frac{5}{104} a^{8} + \frac{19}{78} a^{7} - \frac{49}{208} a^{6} + \frac{17}{52} a^{5} - \frac{119}{312} a^{4} - \frac{227}{936} a^{3} + \frac{139}{624} a^{2} + \frac{35}{117} a - \frac{5}{144}$, $\frac{1}{5616} a^{13} - \frac{1}{5616} a^{12} - \frac{1}{208} a^{11} - \frac{1}{5616} a^{10} - \frac{7}{2808} a^{9} + \frac{3}{52} a^{8} - \frac{133}{624} a^{7} - \frac{29}{624} a^{6} + \frac{3}{52} a^{5} + \frac{799}{2808} a^{4} - \frac{2705}{5616} a^{3} + \frac{51}{208} a^{2} - \frac{275}{5616} a - \frac{49}{432}$, $\frac{1}{5616} a^{14} - \frac{1}{5616} a^{12} + \frac{19}{2808} a^{11} + \frac{1}{468} a^{10} + \frac{4}{351} a^{9} - \frac{29}{624} a^{8} + \frac{3}{52} a^{7} - \frac{1}{39} a^{6} - \frac{137}{2808} a^{5} - \frac{1}{208} a^{4} + \frac{55}{1404} a^{3} + \frac{1909}{5616} a^{2} - \frac{55}{117} a + \frac{49}{216}$, $\frac{1}{196560} a^{15} + \frac{1}{196560} a^{14} + \frac{1}{19656} a^{13} - \frac{37}{196560} a^{12} + \frac{17}{28080} a^{11} + \frac{391}{98280} a^{10} - \frac{619}{65520} a^{9} - \frac{241}{21840} a^{8} + \frac{3919}{21840} a^{7} + \frac{13183}{98280} a^{6} - \frac{11177}{39312} a^{5} + \frac{21803}{196560} a^{4} - \frac{39643}{98280} a^{3} - \frac{3931}{39312} a^{2} + \frac{10349}{39312} a - \frac{23}{126}$, $\frac{1}{982800} a^{16} + \frac{1}{982800} a^{15} + \frac{1}{12285} a^{14} - \frac{1}{491400} a^{13} + \frac{17}{140400} a^{12} + \frac{3967}{982800} a^{11} - \frac{6407}{982800} a^{10} - \frac{4363}{327600} a^{9} - \frac{4271}{109200} a^{8} - \frac{51439}{982800} a^{7} - \frac{925}{39312} a^{6} - \frac{396797}{982800} a^{5} + \frac{191497}{491400} a^{4} - \frac{475}{19656} a^{3} - \frac{941}{15120} a^{2} + \frac{32311}{196560} a - \frac{7}{72}$, $\frac{1}{148804765200} a^{17} - \frac{62329}{148804765200} a^{16} - \frac{1819}{4960158840} a^{15} + \frac{1394}{1328613975} a^{14} + \frac{59}{7570450} a^{13} - \frac{22649813}{148804765200} a^{12} + \frac{20961919}{12400397100} a^{11} - \frac{909619169}{148804765200} a^{10} + \frac{1549519721}{148804765200} a^{9} + \frac{27638809}{5131198800} a^{8} - \frac{44927153}{212578236} a^{7} + \frac{6852511}{49601588400} a^{6} - \frac{28318617973}{74402382600} a^{5} - \frac{94312601}{236198040} a^{4} + \frac{1565183149}{7440238260} a^{3} + \frac{2059943677}{9920317680} a^{2} - \frac{2078291809}{5952190608} a - \frac{14458441}{57232602}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (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:  \( \frac{250282}{9300297825} a^{17} - \frac{1520189}{9300297825} a^{16} + \frac{4653784}{9300297825} a^{15} + \frac{393376}{9300297825} a^{14} - \frac{129932}{28616301} a^{13} + \frac{27733556}{1860059565} a^{12} - \frac{200858576}{9300297825} a^{11} + \frac{31969864}{9300297825} a^{10} + \frac{437389606}{9300297825} a^{9} - \frac{20065646}{320699925} a^{8} - \frac{181120048}{1328613975} a^{7} + \frac{5721582416}{9300297825} a^{6} - \frac{2771723516}{1860059565} a^{5} + \frac{28044774536}{9300297825} a^{4} - \frac{1601341936}{372011913} a^{3} + \frac{838196936}{372011913} a^{2} + \frac{2619999154}{1860059565} a - \frac{1336363}{4088043} \) (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:  \( 20331620.6317 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.24300.1 x3, 3.1.972.1 x3, 3.1.24300.4 x3, 3.1.675.1 x3, 6.0.1771470000.3, 6.0.2834352.1, 6.0.1771470000.2, 6.0.1366875.1, 9.1.387420489000000.14 x9

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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$