↑

Properties

Label 18.0.42192258547...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $23.34$
Ramified primes $2, 3, 5, 11$
Class number $1$
Class group Trivial
Galois group $C_3^2:S_3$ (as 18T24)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -125, 500, -565, 555, -825, 1011, -1054, 761, -675, 690, -334, 226, -159, 70, -30, 11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 11*x^16 - 30*x^15 + 70*x^14 - 159*x^13 + 226*x^12 - 334*x^11 + 690*x^10 - 675*x^9 + 761*x^8 - 1054*x^7 + 1011*x^6 - 825*x^5 + 555*x^4 - 565*x^3 + 500*x^2 - 125*x + 25)
 
gp: K = bnfinit(x^18 - 4*x^17 + 11*x^16 - 30*x^15 + 70*x^14 - 159*x^13 + 226*x^12 - 334*x^11 + 690*x^10 - 675*x^9 + 761*x^8 - 1054*x^7 + 1011*x^6 - 825*x^5 + 555*x^4 - 565*x^3 + 500*x^2 - 125*x + 25, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 11 x^{16} - 30 x^{15} + 70 x^{14} - 159 x^{13} + 226 x^{12} - 334 x^{11} + 690 x^{10} - 675 x^{9} + 761 x^{8} - 1054 x^{7} + 1011 x^{6} - 825 x^{5} + 555 x^{4} - 565 x^{3} + 500 x^{2} - 125 x + 25 \)

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:  \(-4219225854723000000000000=-\,2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.34$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{15} a^{12} + \frac{4}{15} a^{8} + \frac{1}{5} a^{7} + \frac{4}{15} a^{6} + \frac{1}{5} a^{5} + \frac{4}{15} a^{4} - \frac{1}{5} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{15} a^{13} + \frac{1}{15} a^{9} - \frac{2}{5} a^{8} + \frac{1}{15} a^{7} - \frac{1}{5} a^{6} + \frac{1}{15} a^{5} + \frac{1}{5} a^{4} + \frac{2}{15} a^{3} - \frac{1}{3} a$, $\frac{1}{15} a^{14} + \frac{1}{15} a^{10} + \frac{4}{15} a^{8} + \frac{1}{5} a^{7} - \frac{2}{15} a^{6} - \frac{2}{5} a^{5} + \frac{1}{3} a^{4} + \frac{2}{5} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{105} a^{15} + \frac{1}{35} a^{14} - \frac{1}{35} a^{13} + \frac{1}{35} a^{12} + \frac{1}{105} a^{11} - \frac{2}{35} a^{10} + \frac{2}{21} a^{9} - \frac{1}{5} a^{8} + \frac{16}{105} a^{7} + \frac{4}{35} a^{6} + \frac{47}{105} a^{5} + \frac{16}{35} a^{4} + \frac{7}{15} a^{3} - \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{3045} a^{16} - \frac{4}{1015} a^{15} + \frac{43}{3045} a^{14} - \frac{33}{1015} a^{13} + \frac{82}{3045} a^{12} - \frac{1}{29} a^{11} + \frac{23}{3045} a^{10} + \frac{76}{1015} a^{9} - \frac{1069}{3045} a^{8} + \frac{246}{1015} a^{7} - \frac{32}{145} a^{6} - \frac{422}{1015} a^{5} + \frac{8}{203} a^{4} + \frac{284}{1015} a^{3} - \frac{10}{609} a^{2} + \frac{89}{203} a + \frac{47}{203}$, $\frac{1}{99596560152075} a^{17} - \frac{5711182124}{99596560152075} a^{16} - \frac{127017716608}{33198853384025} a^{15} + \frac{25572710326}{2845616004345} a^{14} - \frac{126246376748}{19919312030415} a^{13} + \frac{2101712829116}{99596560152075} a^{12} - \frac{6792964570124}{99596560152075} a^{11} - \frac{517054540273}{33198853384025} a^{10} + \frac{1914391000498}{19919312030415} a^{9} + \frac{606030308341}{3983862406083} a^{8} + \frac{14403480693722}{33198853384025} a^{7} - \frac{34685816497639}{99596560152075} a^{6} - \frac{23533058229049}{99596560152075} a^{5} - \frac{376315836086}{6639770676805} a^{4} - \frac{51753781253}{137374565727} a^{3} - \frac{2189026277104}{19919312030415} a^{2} - \frac{1556952554459}{3983862406083} a + \frac{1447913601466}{3983862406083}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{32006120422}{3434364143175} a^{17} + \frac{17879358509}{490623449025} a^{16} - \frac{330633862567}{3434364143175} a^{15} + \frac{8534206084}{32708229935} a^{14} - \frac{413846073481}{686872828635} a^{13} + \frac{222239512673}{163541149675} a^{12} - \frac{6305627509547}{3434364143175} a^{11} + \frac{8978669096723}{3434364143175} a^{10} - \frac{1340922602003}{228957609545} a^{9} + \frac{1207426530719}{228957609545} a^{8} - \frac{18998737944307}{3434364143175} a^{7} + \frac{30730050637688}{3434364143175} a^{6} - \frac{26890207458742}{3434364143175} a^{5} + \frac{1563701925374}{228957609545} a^{4} - \frac{189890616446}{45791521909} a^{3} + \frac{3222773415943}{686872828635} a^{2} - \frac{679958525065}{137374565727} a + \frac{172518070402}{137374565727} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 163709.22551032295 \)
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

\(\Q(\sqrt{-3}) \), 3.1.300.1 x3, 6.0.270000.1, 9.3.395307000000.1

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 R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$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}$
$11$11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$