↑

Properties

Label 18.0.85493948743...2347.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 7^{12}$
Root discriminant $24.27$
Ramified primes $3, 7$
Class number $3$
Class group $[3]$
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![811, -5190, 14379, -20937, 16131, -4305, -4059, 7515, -5535, 2354, 264, -1407, 1359, -861, 411, -156, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 411*x^14 - 861*x^13 + 1359*x^12 - 1407*x^11 + 264*x^10 + 2354*x^9 - 5535*x^8 + 7515*x^7 - 4059*x^6 - 4305*x^5 + 16131*x^4 - 20937*x^3 + 14379*x^2 - 5190*x + 811)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 411*x^14 - 861*x^13 + 1359*x^12 - 1407*x^11 + 264*x^10 + 2354*x^9 - 5535*x^8 + 7515*x^7 - 4059*x^6 - 4305*x^5 + 16131*x^4 - 20937*x^3 + 14379*x^2 - 5190*x + 811, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 411 x^{14} - 861 x^{13} + 1359 x^{12} - 1407 x^{11} + 264 x^{10} + 2354 x^{9} - 5535 x^{8} + 7515 x^{7} - 4059 x^{6} - 4305 x^{5} + 16131 x^{4} - 20937 x^{3} + 14379 x^{2} - 5190 x + 811 \)

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:  \(-8549394874383196572862347=-\,3^{31}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.27$
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:  $3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{8} + \frac{1}{10} a^{7} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{10} a^{9} - \frac{1}{10} a^{7} - \frac{1}{10} a^{6} - \frac{2}{5} a^{5} - \frac{3}{10} a^{4} + \frac{3}{10} a^{3} - \frac{3}{10} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{10} + \frac{1}{10} a^{6} + \frac{1}{10} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{10}$, $\frac{1}{10} a^{11} + \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{10} a$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{10} - \frac{3}{20} a^{6} + \frac{1}{5} a^{5} + \frac{9}{20} a^{4} - \frac{2}{5} a^{3} + \frac{1}{10} a^{2} - \frac{3}{20}$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{11} - \frac{3}{20} a^{7} + \frac{1}{5} a^{6} + \frac{9}{20} a^{5} - \frac{2}{5} a^{4} + \frac{1}{10} a^{3} - \frac{3}{20} a$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{10} - \frac{1}{20} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{3}{10} a^{5} - \frac{9}{20} a^{4} + \frac{3}{10} a^{3} + \frac{9}{20} a^{2} - \frac{3}{10} a + \frac{1}{4}$, $\frac{1}{220} a^{15} - \frac{1}{110} a^{14} + \frac{1}{55} a^{13} - \frac{3}{220} a^{12} - \frac{3}{220} a^{11} - \frac{1}{20} a^{10} - \frac{1}{44} a^{9} + \frac{3}{110} a^{8} - \frac{27}{110} a^{7} + \frac{7}{220} a^{6} + \frac{21}{44} a^{5} - \frac{63}{220} a^{4} + \frac{103}{220} a^{3} - \frac{26}{55} a^{2} - \frac{57}{220} a + \frac{93}{220}$, $\frac{1}{1210003300} a^{16} - \frac{2}{302500825} a^{15} + \frac{5686344}{302500825} a^{14} + \frac{22283003}{1210003300} a^{13} + \frac{32902}{302500825} a^{12} - \frac{48468391}{1210003300} a^{11} - \frac{705579}{121000330} a^{10} + \frac{5620591}{302500825} a^{9} + \frac{27354359}{605001650} a^{8} - \frac{189727097}{1210003300} a^{7} + \frac{2771079}{121000330} a^{6} + \frac{13066167}{110000300} a^{5} + \frac{63315067}{302500825} a^{4} + \frac{56874853}{605001650} a^{3} - \frac{380730861}{1210003300} a^{2} - \frac{95317689}{1210003300} a + \frac{419206351}{1210003300}$, $\frac{1}{219010597300} a^{17} + \frac{41}{109505298650} a^{16} + \frac{88186389}{54752649325} a^{15} - \frac{293570201}{109505298650} a^{14} - \frac{2663910857}{219010597300} a^{13} + \frac{2119382209}{219010597300} a^{12} + \frac{319661059}{43802119460} a^{11} - \frac{5392051771}{219010597300} a^{10} - \frac{3536951741}{109505298650} a^{9} - \frac{545229181}{109505298650} a^{8} + \frac{8115185843}{43802119460} a^{7} - \frac{38612413563}{219010597300} a^{6} + \frac{21279287663}{219010597300} a^{5} + \frac{88208851921}{219010597300} a^{4} + \frac{64658892139}{219010597300} a^{3} - \frac{24146316587}{109505298650} a^{2} - \frac{26299037576}{54752649325} a + \frac{12074426669}{43802119460}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( \frac{142478274}{54752649325} a^{17} - \frac{1211065329}{54752649325} a^{16} + \frac{1164716012}{10950529865} a^{15} - \frac{3891108774}{10950529865} a^{14} + \frac{49458012006}{54752649325} a^{13} - \frac{19974525194}{10950529865} a^{12} + \frac{148166499222}{54752649325} a^{11} - \frac{12233974344}{4977513575} a^{10} - \frac{19546183936}{54752649325} a^{9} + \frac{320886743151}{54752649325} a^{8} - \frac{640182526626}{54752649325} a^{7} + \frac{785562146878}{54752649325} a^{6} - \frac{237498764142}{54752649325} a^{5} - \frac{678971147157}{54752649325} a^{4} + \frac{1973970226214}{54752649325} a^{3} - \frac{189472903824}{4977513575} a^{2} + \frac{1171992574812}{54752649325} a - \frac{242868047207}{54752649325} \) (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:  \( 102362.355108 \)
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

\(\Q(\sqrt{-3}) \), 3.1.1323.1 x3, 3.3.3969.2, 6.0.5250987.1, 6.0.964467.2 x2, 6.0.47258883.1, 9.3.1688134559643.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.964467.2
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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$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}$