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Properties

Label 18.0.45028390589...0000.3
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![1444, -4788, 9036, -9228, 3600, 5760, -10983, 9243, -3177, -1364, 2601, -1989, 1200, -711, 369, -150, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 150*x^15 + 369*x^14 - 711*x^13 + 1200*x^12 - 1989*x^11 + 2601*x^10 - 1364*x^9 - 3177*x^8 + 9243*x^7 - 10983*x^6 + 5760*x^5 + 3600*x^4 - 9228*x^3 + 9036*x^2 - 4788*x + 1444)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 150*x^15 + 369*x^14 - 711*x^13 + 1200*x^12 - 1989*x^11 + 2601*x^10 - 1364*x^9 - 3177*x^8 + 9243*x^7 - 10983*x^6 + 5760*x^5 + 3600*x^4 - 9228*x^3 + 9036*x^2 - 4788*x + 1444, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 150 x^{15} + 369 x^{14} - 711 x^{13} + 1200 x^{12} - 1989 x^{11} + 2601 x^{10} - 1364 x^{9} - 3177 x^{8} + 9243 x^{7} - 10983 x^{6} + 5760 x^{5} + 3600 x^{4} - 9228 x^{3} + 9036 x^{2} - 4788 x + 1444 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{2}{9}$, $\frac{1}{9} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{2}{9} a$, $\frac{1}{9} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{2}{9} a^{2}$, $\frac{1}{126} a^{12} - \frac{1}{21} a^{11} - \frac{1}{126} a^{9} - \frac{2}{7} a^{8} - \frac{5}{14} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{59}{126} a^{3} - \frac{2}{21} a^{2} - \frac{2}{7} a + \frac{11}{63}$, $\frac{1}{756} a^{13} - \frac{1}{756} a^{12} - \frac{5}{126} a^{11} + \frac{41}{756} a^{10} + \frac{1}{756} a^{9} - \frac{5}{21} a^{8} - \frac{5}{84} a^{7} + \frac{37}{84} a^{6} + \frac{17}{42} a^{5} - \frac{13}{756} a^{4} + \frac{31}{756} a^{3} + \frac{13}{63} a^{2} + \frac{89}{378} a - \frac{29}{378}$, $\frac{1}{756} a^{14} - \frac{1}{756} a^{12} - \frac{1}{756} a^{11} - \frac{1}{18} a^{10} - \frac{41}{756} a^{9} - \frac{11}{28} a^{8} - \frac{2}{7} a^{7} + \frac{11}{28} a^{6} - \frac{103}{756} a^{5} - \frac{3}{14} a^{4} - \frac{59}{756} a^{3} + \frac{155}{378} a^{2} - \frac{31}{63} a + \frac{13}{54}$, $\frac{1}{143640} a^{15} + \frac{13}{47880} a^{14} - \frac{41}{143640} a^{13} + \frac{11}{23940} a^{12} + \frac{2453}{47880} a^{11} + \frac{2057}{143640} a^{10} + \frac{65}{2052} a^{9} - \frac{3659}{15960} a^{8} + \frac{845}{3192} a^{7} - \frac{21683}{71820} a^{6} - \frac{11677}{47880} a^{5} + \frac{53723}{143640} a^{4} - \frac{1039}{47880} a^{3} + \frac{1229}{23940} a^{2} - \frac{30727}{71820} a - \frac{523}{3780}$, $\frac{1}{1436400} a^{16} + \frac{1}{478800} a^{15} + \frac{43}{95760} a^{14} - \frac{58}{89775} a^{13} - \frac{211}{205200} a^{12} - \frac{23339}{478800} a^{11} + \frac{4}{75} a^{10} - \frac{70571}{1436400} a^{9} + \frac{53299}{159600} a^{8} - \frac{43129}{359100} a^{7} - \frac{18869}{95760} a^{6} - \frac{136217}{478800} a^{5} + \frac{1411}{3024} a^{4} - \frac{1459}{5400} a^{3} + \frac{29557}{239400} a^{2} + \frac{317}{760} a - \frac{503}{9450}$, $\frac{1}{33037200} a^{17} + \frac{1}{5506200} a^{16} - \frac{13}{16518600} a^{15} + \frac{14387}{33037200} a^{14} + \frac{2017}{4719600} a^{13} + \frac{4892}{2064825} a^{12} - \frac{384421}{11012400} a^{11} + \frac{46793}{33037200} a^{10} + \frac{1711}{41400} a^{9} + \frac{10099337}{33037200} a^{8} - \frac{213607}{478800} a^{7} - \frac{2911519}{8259300} a^{6} - \frac{1717637}{8259300} a^{5} - \frac{15804559}{33037200} a^{4} + \frac{477101}{1651860} a^{3} - \frac{947747}{2753100} a^{2} - \frac{950839}{2359800} a - \frac{3769}{10350}$

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:  \( \frac{10667}{4719600} a^{17} - \frac{6811}{471960} a^{16} + \frac{127531}{2359800} a^{15} - \frac{30529}{248400} a^{14} + \frac{896969}{4719600} a^{13} - \frac{121439}{589950} a^{12} + \frac{1553543}{4719600} a^{11} - \frac{659237}{943920} a^{10} - \frac{1350161}{2359800} a^{9} + \frac{16292647}{4719600} a^{8} - \frac{877003}{205200} a^{7} + \frac{612097}{1179900} a^{6} + \frac{4181657}{589950} a^{5} - \frac{28677833}{4719600} a^{4} + \frac{1611161}{589950} a^{3} + \frac{4760767}{1179900} a^{2} - \frac{7193551}{2359800} a + \frac{45049}{15525} \) (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:  \( 13593134.6098 \) (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

\(\Q(\sqrt{-3}) \), 3.1.6075.1 x3, 3.1.972.1 x3, 3.1.2700.1 x3, 3.1.24300.2 x3, 6.0.110716875.1, 6.0.2834352.1, 6.0.21870000.2, 6.0.1771470000.4, 9.1.387420489000000.20 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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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}$