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Properties

Label 18.18.1167818901...7857.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{44}\cdot 17^{9}$
Root discriminant $60.47$
Ramified primes $3, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2161, 14355, -2259, -131820, 192555, 186741, -455655, 35910, 292437, -91106, -76374, 33174, 8472, -5040, -252, 348, -18, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 18*x^16 + 348*x^15 - 252*x^14 - 5040*x^13 + 8472*x^12 + 33174*x^11 - 76374*x^10 - 91106*x^9 + 292437*x^8 + 35910*x^7 - 455655*x^6 + 186741*x^5 + 192555*x^4 - 131820*x^3 - 2259*x^2 + 14355*x - 2161)
 
gp: K = bnfinit(x^18 - 9*x^17 - 18*x^16 + 348*x^15 - 252*x^14 - 5040*x^13 + 8472*x^12 + 33174*x^11 - 76374*x^10 - 91106*x^9 + 292437*x^8 + 35910*x^7 - 455655*x^6 + 186741*x^5 + 192555*x^4 - 131820*x^3 - 2259*x^2 + 14355*x - 2161, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 18 x^{16} + 348 x^{15} - 252 x^{14} - 5040 x^{13} + 8472 x^{12} + 33174 x^{11} - 76374 x^{10} - 91106 x^{9} + 292437 x^{8} + 35910 x^{7} - 455655 x^{6} + 186741 x^{5} + 192555 x^{4} - 131820 x^{3} - 2259 x^{2} + 14355 x - 2161 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(116781890125989356502353933497857=3^{44}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.47$
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:  $3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(459=3^{3}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{459}(256,·)$, $\chi_{459}(1,·)$, $\chi_{459}(322,·)$, $\chi_{459}(67,·)$, $\chi_{459}(205,·)$, $\chi_{459}(271,·)$, $\chi_{459}(16,·)$, $\chi_{459}(409,·)$, $\chi_{459}(154,·)$, $\chi_{459}(220,·)$, $\chi_{459}(358,·)$, $\chi_{459}(103,·)$, $\chi_{459}(424,·)$, $\chi_{459}(169,·)$, $\chi_{459}(307,·)$, $\chi_{459}(52,·)$, $\chi_{459}(373,·)$, $\chi_{459}(118,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{54849393073733009857881388686514037953} a^{17} - \frac{19164030115436102146493996814277932546}{54849393073733009857881388686514037953} a^{16} + \frac{4158245565130784695251449092993474130}{54849393073733009857881388686514037953} a^{15} + \frac{4980102323197351160890088531419283765}{54849393073733009857881388686514037953} a^{14} + \frac{13903335595864738832058225317079646038}{54849393073733009857881388686514037953} a^{13} - \frac{14579023142590663681017942404045031745}{54849393073733009857881388686514037953} a^{12} - \frac{19238541068614478775899961970114092220}{54849393073733009857881388686514037953} a^{11} - \frac{24091417621117095384427946744081486291}{54849393073733009857881388686514037953} a^{10} - \frac{10438349140660199589905713260896326471}{54849393073733009857881388686514037953} a^{9} + \frac{6305685268101957909480542434924073581}{54849393073733009857881388686514037953} a^{8} + \frac{25712548691560364591234965015684201957}{54849393073733009857881388686514037953} a^{7} + \frac{23001880382961631392310176691655811279}{54849393073733009857881388686514037953} a^{6} + \frac{1759900813994170052844989395952421206}{54849393073733009857881388686514037953} a^{5} - \frac{21963375347337508318406839842774113537}{54849393073733009857881388686514037953} a^{4} + \frac{24062603502077695068193126800062029797}{54849393073733009857881388686514037953} a^{3} - \frac{741798429812861955411700078985875157}{54849393073733009857881388686514037953} a^{2} - \frac{20713704974143220779804113440095302712}{54849393073733009857881388686514037953} a + \frac{21740095684248024877264943215937036853}{54849393073733009857881388686514037953}$

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:  $17$
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:  \( 13605878233.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{9})^+\), 6.6.32234193.1, \(\Q(\zeta_{27})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ $18$ $18$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ $18$ $18$ $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$17$17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$