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Properties

Label 18.0.36542633964...0208.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 31^{8}$
Root discriminant $26.31$
Ramified primes $2, 3, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 9, 54, 203, 579, 1041, 1559, 1506, 1869, 1455, 1320, 552, 312, 21, 42, -2, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 2*x^15 + 42*x^14 + 21*x^13 + 312*x^12 + 552*x^11 + 1320*x^10 + 1455*x^9 + 1869*x^8 + 1506*x^7 + 1559*x^6 + 1041*x^5 + 579*x^4 + 203*x^3 + 54*x^2 + 9*x + 1)
 
gp: K = bnfinit(x^18 + 9*x^16 - 2*x^15 + 42*x^14 + 21*x^13 + 312*x^12 + 552*x^11 + 1320*x^10 + 1455*x^9 + 1869*x^8 + 1506*x^7 + 1559*x^6 + 1041*x^5 + 579*x^4 + 203*x^3 + 54*x^2 + 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - 2 x^{15} + 42 x^{14} + 21 x^{13} + 312 x^{12} + 552 x^{11} + 1320 x^{10} + 1455 x^{9} + 1869 x^{8} + 1506 x^{7} + 1559 x^{6} + 1041 x^{5} + 579 x^{4} + 203 x^{3} + 54 x^{2} + 9 x + 1 \)

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:  \(-36542633964773200992350208=-\,2^{12}\cdot 3^{21}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.31$
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:  $2, 3, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6}$, $\frac{1}{30} a^{13} + \frac{1}{30} a^{12} - \frac{1}{15} a^{11} - \frac{2}{15} a^{10} - \frac{1}{10} a^{9} - \frac{1}{30} a^{8} - \frac{4}{15} a^{7} - \frac{11}{30} a^{5} + \frac{13}{30} a^{4} + \frac{11}{30} a^{3} + \frac{7}{30} a^{2} - \frac{1}{10} a + \frac{1}{30}$, $\frac{1}{30} a^{14} + \frac{1}{15} a^{12} - \frac{1}{15} a^{11} + \frac{1}{30} a^{10} + \frac{1}{15} a^{9} + \frac{4}{15} a^{8} + \frac{4}{15} a^{7} - \frac{11}{30} a^{6} - \frac{1}{5} a^{5} + \frac{13}{30} a^{4} + \frac{1}{5} a^{3} + \frac{1}{6} a^{2} + \frac{2}{15} a + \frac{2}{15}$, $\frac{1}{19530} a^{15} + \frac{2}{1395} a^{14} - \frac{44}{3255} a^{13} - \frac{98}{1395} a^{12} + \frac{61}{2790} a^{11} - \frac{1013}{9765} a^{10} - \frac{929}{9765} a^{9} + \frac{1564}{9765} a^{8} - \frac{3719}{19530} a^{7} + \frac{1}{155} a^{6} + \frac{1117}{6510} a^{5} - \frac{4184}{9765} a^{4} + \frac{4177}{19530} a^{3} - \frac{2789}{9765} a^{2} + \frac{4777}{9765} a - \frac{2557}{9765}$, $\frac{1}{19530} a^{16} + \frac{127}{9765} a^{14} + \frac{23}{2790} a^{13} - \frac{1}{90} a^{12} + \frac{821}{9765} a^{11} + \frac{8}{105} a^{10} - \frac{929}{6510} a^{9} + \frac{6347}{19530} a^{8} + \frac{379}{1395} a^{7} + \frac{559}{2170} a^{6} + \frac{11}{19530} a^{5} - \frac{1259}{3906} a^{4} + \frac{3109}{19530} a^{3} - \frac{97}{217} a^{2} + \frac{4049}{19530} a + \frac{649}{1395}$, $\frac{1}{46539990} a^{17} - \frac{101}{6648570} a^{16} - \frac{286}{23269995} a^{15} - \frac{2956}{369365} a^{14} - \frac{23818}{3324285} a^{13} + \frac{956972}{23269995} a^{12} + \frac{206503}{7756665} a^{11} + \frac{283663}{4653999} a^{10} + \frac{155081}{2585555} a^{9} + \frac{253672}{3324285} a^{8} - \frac{12921029}{46539990} a^{7} + \frac{8279009}{46539990} a^{6} - \frac{783506}{4653999} a^{5} - \frac{4571731}{23269995} a^{4} - \frac{2053377}{5171110} a^{3} + \frac{2170047}{5171110} a^{2} - \frac{736897}{6648570} a - \frac{2803007}{6648570}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{1722052}{7756665} a^{17} - \frac{96703}{2585555} a^{16} + \frac{5195503}{2585555} a^{15} - \frac{6075176}{7756665} a^{14} + \frac{73717109}{7756665} a^{13} + \frac{23615401}{7756665} a^{12} + \frac{535088513}{7756665} a^{11} + \frac{286987643}{2585555} a^{10} + \frac{713593287}{2585555} a^{9} + \frac{103096931}{369365} a^{8} + \frac{969027037}{2585555} a^{7} + \frac{102885804}{369365} a^{6} + \frac{343305278}{1108095} a^{5} + \frac{1450217068}{7756665} a^{4} + \frac{821262812}{7756665} a^{3} + \frac{82273839}{2585555} a^{2} + \frac{78170927}{7756665} a + \frac{13123018}{7756665} \) (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:  \( 637987.1635985387 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

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 $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.34992.1, 9.3.1163370485952.3

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

Sibling fields

Degree 9 siblings: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
3Data not computed
$31$31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$