↑

Properties

Label 18.0.63472850187...9123.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{12}\cdot 13^{12}$
Root discriminant $35.04$
Ramified primes $3, 7, 13$
Class number $36$ (GRH)
Class group $[6, 6]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, -675, 6264, -15975, 19080, -10974, 535, 417, 4383, -5715, 4146, -2067, 832, -399, 225, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 225*x^14 - 399*x^13 + 832*x^12 - 2067*x^11 + 4146*x^10 - 5715*x^9 + 4383*x^8 + 417*x^7 + 535*x^6 - 10974*x^5 + 19080*x^4 - 15975*x^3 + 6264*x^2 - 675*x + 27)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 225*x^14 - 399*x^13 + 832*x^12 - 2067*x^11 + 4146*x^10 - 5715*x^9 + 4383*x^8 + 417*x^7 + 535*x^6 - 10974*x^5 + 19080*x^4 - 15975*x^3 + 6264*x^2 - 675*x + 27, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 225 x^{14} - 399 x^{13} + 832 x^{12} - 2067 x^{11} + 4146 x^{10} - 5715 x^{9} + 4383 x^{8} + 417 x^{7} + 535 x^{6} - 10974 x^{5} + 19080 x^{4} - 15975 x^{3} + 6264 x^{2} - 675 x + 27 \)

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:  \(-6347285018761982937208599123=-\,3^{9}\cdot 7^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.04$
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:  $3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{5}$, $\frac{1}{27} a^{12} - \frac{1}{27} a^{10} - \frac{1}{9} a^{7} - \frac{4}{27} a^{6} - \frac{4}{9} a^{5} + \frac{7}{27} a^{4} - \frac{4}{9} a^{3} + \frac{2}{9} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{11} - \frac{1}{9} a^{8} - \frac{4}{27} a^{7} - \frac{1}{9} a^{6} + \frac{7}{27} a^{5} - \frac{1}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{297} a^{14} + \frac{4}{297} a^{13} + \frac{5}{297} a^{12} - \frac{16}{297} a^{11} - \frac{4}{99} a^{10} - \frac{2}{99} a^{9} + \frac{38}{297} a^{8} - \frac{28}{297} a^{7} - \frac{2}{297} a^{6} + \frac{37}{297} a^{5} + \frac{41}{99} a^{4} + \frac{5}{11} a^{3} - \frac{13}{33} a^{2} - \frac{7}{33} a - \frac{2}{11}$, $\frac{1}{297} a^{15} - \frac{1}{99} a^{12} + \frac{8}{297} a^{11} + \frac{1}{33} a^{10} - \frac{4}{297} a^{9} - \frac{5}{99} a^{8} - \frac{1}{9} a^{7} - \frac{7}{99} a^{6} - \frac{113}{297} a^{5} - \frac{20}{99} a^{4} - \frac{43}{99} a^{3} - \frac{10}{33} a^{2} - \frac{1}{3} a - \frac{3}{11}$, $\frac{1}{560136951} a^{16} - \frac{8}{560136951} a^{15} - \frac{745190}{560136951} a^{14} + \frac{5216470}{560136951} a^{13} - \frac{8214512}{560136951} a^{12} - \frac{18527402}{560136951} a^{11} - \frac{24687832}{560136951} a^{10} - \frac{18061573}{560136951} a^{9} + \frac{8671346}{560136951} a^{8} + \frac{30175490}{560136951} a^{7} - \frac{50141101}{560136951} a^{6} + \frac{172485506}{560136951} a^{5} - \frac{10051280}{186712317} a^{4} - \frac{3691384}{16973847} a^{3} - \frac{697990}{62237439} a^{2} + \frac{627073}{5657949} a + \frac{1205579}{6915271}$, $\frac{1}{208931082723} a^{17} + \frac{178}{208931082723} a^{16} - \frac{36580355}{208931082723} a^{15} - \frac{216372122}{208931082723} a^{14} + \frac{339674518}{208931082723} a^{13} - \frac{2808149261}{208931082723} a^{12} - \frac{8238549628}{208931082723} a^{11} + \frac{751851344}{208931082723} a^{10} + \frac{8182004813}{208931082723} a^{9} - \frac{10557378181}{208931082723} a^{8} - \frac{4271015323}{208931082723} a^{7} - \frac{26699059129}{208931082723} a^{6} - \frac{9999077293}{23214564747} a^{5} - \frac{7529772746}{23214564747} a^{4} - \frac{5517563509}{23214564747} a^{3} - \frac{2484790973}{7738188249} a^{2} + \frac{1364842108}{7738188249} a - \frac{2442302173}{7738188249}$

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

Class group and class number

$C_{6}\times C_{6}$, which has order $36$ (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:  \( \frac{273282}{21317323} a^{17} - \frac{2322897}{21317323} a^{16} + \frac{257657692}{575567721} a^{15} - \frac{226022770}{191855907} a^{14} + \frac{1360044094}{575567721} a^{13} - \frac{787051954}{191855907} a^{12} + \frac{5145629038}{575567721} a^{11} - \frac{4331616916}{191855907} a^{10} + \frac{24822274466}{575567721} a^{9} - \frac{10540719332}{191855907} a^{8} + \frac{19882022810}{575567721} a^{7} + \frac{3046293190}{191855907} a^{6} + \frac{10170609104}{575567721} a^{5} - \frac{24505195901}{191855907} a^{4} + \frac{35562483478}{191855907} a^{3} - \frac{2792178854}{21317323} a^{2} + \frac{2405792693}{63951969} a - \frac{35650823}{21317323} \) (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:  \( 5151205.65339 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.24843.1 x3, 3.3.8281.2, 6.0.1851523947.3, 6.0.1851523947.1, 6.0.223587.1 x2, 9.3.15332469805107.3 x3

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

Sibling fields

Degree 6 sibling: 6.0.223587.1
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$