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Properties

Label 18.0.21625835374...8928.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 83^{9}$
Root discriminant $62.57$
Ramified primes $2, 3, 83$
Class number $1008$ (GRH)
Class group $[2, 2, 2, 126]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1259712, 0, 451008, 188640, 590976, -303840, 440208, -208800, 171936, -58400, 48168, -11160, 6276, -1440, 468, -60, 36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 36*x^16 - 60*x^15 + 468*x^14 - 1440*x^13 + 6276*x^12 - 11160*x^11 + 48168*x^10 - 58400*x^9 + 171936*x^8 - 208800*x^7 + 440208*x^6 - 303840*x^5 + 590976*x^4 + 188640*x^3 + 451008*x^2 + 1259712)
 
gp: K = bnfinit(x^18 + 36*x^16 - 60*x^15 + 468*x^14 - 1440*x^13 + 6276*x^12 - 11160*x^11 + 48168*x^10 - 58400*x^9 + 171936*x^8 - 208800*x^7 + 440208*x^6 - 303840*x^5 + 590976*x^4 + 188640*x^3 + 451008*x^2 + 1259712, 1)
 

Normalized defining polynomial

\( x^{18} + 36 x^{16} - 60 x^{15} + 468 x^{14} - 1440 x^{13} + 6276 x^{12} - 11160 x^{11} + 48168 x^{10} - 58400 x^{9} + 171936 x^{8} - 208800 x^{7} + 440208 x^{6} - 303840 x^{5} + 590976 x^{4} + 188640 x^{3} + 451008 x^{2} + 1259712 \)

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:  \(-216258353745310739104685841788928=-\,2^{12}\cdot 3^{24}\cdot 83^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.57$
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:  $2, 3, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{12} a^{6} - \frac{1}{6} a^{3}$, $\frac{1}{12} a^{7} - \frac{1}{6} a^{4}$, $\frac{1}{12} a^{8} - \frac{1}{6} a^{5}$, $\frac{1}{24} a^{9} - \frac{1}{6} a^{3}$, $\frac{1}{24} a^{10} - \frac{1}{6} a^{4}$, $\frac{1}{24} a^{11} - \frac{1}{6} a^{5}$, $\frac{1}{11088} a^{12} + \frac{5}{1848} a^{11} + \frac{1}{616} a^{10} + \frac{109}{5544} a^{9} + \frac{1}{462} a^{8} + \frac{1}{42} a^{7} - \frac{1}{252} a^{6} + \frac{52}{231} a^{5} + \frac{5}{21} a^{4} + \frac{23}{154} a^{3} - \frac{34}{77} a^{2} - \frac{8}{77}$, $\frac{1}{11088} a^{13} + \frac{1}{264} a^{11} + \frac{5}{396} a^{10} - \frac{1}{231} a^{9} - \frac{19}{462} a^{8} + \frac{2}{63} a^{7} + \frac{5}{462} a^{6} + \frac{5}{33} a^{5} - \frac{37}{231} a^{4} - \frac{41}{462} a^{3} + \frac{19}{77} a^{2} - \frac{8}{77} a + \frac{9}{77}$, $\frac{1}{498960} a^{14} + \frac{1}{49896} a^{13} + \frac{1}{41580} a^{12} + \frac{445}{24948} a^{11} + \frac{289}{35640} a^{10} + \frac{1}{594} a^{9} + \frac{53}{1620} a^{8} - \frac{107}{12474} a^{7} + \frac{79}{1980} a^{6} + \frac{25}{462} a^{5} - \frac{1}{5} a^{4} - \frac{23}{99} a^{3} + \frac{971}{3465} a^{2} - \frac{122}{693} a + \frac{8}{35}$, $\frac{1}{76839840} a^{15} - \frac{31}{38419920} a^{14} + \frac{467}{12806640} a^{13} + \frac{127}{9604980} a^{12} + \frac{277199}{19209960} a^{11} - \frac{87341}{6403320} a^{10} + \frac{21631}{19209960} a^{9} - \frac{158303}{4802490} a^{8} - \frac{7214}{266805} a^{7} + \frac{10729}{355740} a^{6} - \frac{18551}{177870} a^{5} - \frac{2426}{38115} a^{4} + \frac{3881}{533610} a^{3} + \frac{6544}{266805} a^{2} - \frac{4}{55} a + \frac{9022}{29645}$, $\frac{1}{3457792800} a^{16} + \frac{1}{691558560} a^{15} + \frac{17}{1728896400} a^{14} + \frac{2389}{69155856} a^{13} + \frac{13}{898128} a^{12} - \frac{321694}{21611205} a^{11} - \frac{15383531}{864448200} a^{10} - \frac{211745}{17288964} a^{9} + \frac{3070238}{108056025} a^{8} - \frac{1699}{152460} a^{7} + \frac{97897}{8004150} a^{6} - \frac{247}{436590} a^{5} + \frac{68435}{480249} a^{4} - \frac{506414}{2401245} a^{3} - \frac{1603823}{12006225} a^{2} + \frac{8626}{29645} a + \frac{50027}{148225}$, $\frac{1}{12100282395584090400} a^{17} + \frac{1579098343}{12100282395584090400} a^{16} - \frac{17716333561}{12100282395584090400} a^{15} + \frac{3860145091631}{6050141197792045200} a^{14} + \frac{3637144225039}{605014119779204520} a^{13} - \frac{926937790153}{60501411977920452} a^{12} + \frac{39082028223367799}{3025070598896022600} a^{11} + \frac{3149607276193307}{275006418081456600} a^{10} - \frac{1519410639384931}{3025070598896022600} a^{9} + \frac{2806261939891589}{108038235674857950} a^{8} - \frac{769813054914997}{42014869429111425} a^{7} + \frac{4376893581799951}{168059477716445700} a^{6} - \frac{131085161448221}{1680594777164457} a^{5} - \frac{4311535838867}{22865235063462} a^{4} + \frac{8638129578570049}{84029738858222850} a^{3} + \frac{19467302827936991}{42014869429111425} a^{2} - \frac{2288423487467162}{4668318825456825} a - \frac{104840327210104}{518702091717425}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{126}$, which has order $1008$ (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:  \( -1 \) (order $2$)
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:  \( 2929141.96377 \) (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

\(\Q(\sqrt{-83}) \), 3.1.26892.3 x3, \(\Q(\zeta_{9})^+\), 6.0.60023912112.1, 6.0.741035952.5 x2, 6.0.3751494507.2, 9.3.19447747524288.10 x3

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

Sibling fields

Degree 6 sibling: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
$83$83.6.3.2$x^{6} - 6889 x^{2} + 1715361$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
83.6.3.2$x^{6} - 6889 x^{2} + 1715361$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
83.6.3.2$x^{6} - 6889 x^{2} + 1715361$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$