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Properties

Label 18.0.96718860867...4923.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{45}\cdot 41^{9}$
Root discriminant $99.81$
Ramified primes $3, 41$
Class number $806474$ (GRH)
Class group $[19, 42446]$ (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![18077629681, -11051190000, 8100000000, -3683730000, 5400000000, -331535700, 1386000000, -11051190, 178200000, -122791, 12870000, 0, 546000, 0, 13500, 0, 180, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 180*x^16 + 13500*x^14 + 546000*x^12 + 12870000*x^10 - 122791*x^9 + 178200000*x^8 - 11051190*x^7 + 1386000000*x^6 - 331535700*x^5 + 5400000000*x^4 - 3683730000*x^3 + 8100000000*x^2 - 11051190000*x + 18077629681)
 
gp: K = bnfinit(x^18 + 180*x^16 + 13500*x^14 + 546000*x^12 + 12870000*x^10 - 122791*x^9 + 178200000*x^8 - 11051190*x^7 + 1386000000*x^6 - 331535700*x^5 + 5400000000*x^4 - 3683730000*x^3 + 8100000000*x^2 - 11051190000*x + 18077629681, 1)
 

Normalized defining polynomial

\( x^{18} + 180 x^{16} + 13500 x^{14} + 546000 x^{12} + 12870000 x^{10} - 122791 x^{9} + 178200000 x^{8} - 11051190 x^{7} + 1386000000 x^{6} - 331535700 x^{5} + 5400000000 x^{4} - 3683730000 x^{3} + 8100000000 x^{2} - 11051190000 x + 18077629681 \)

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:  \(-967188608675270393759591510313094923=-\,3^{45}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.81$
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:  $3, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1107=3^{3}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{1107}(1,·)$, $\chi_{1107}(1106,·)$, $\chi_{1107}(983,·)$, $\chi_{1107}(985,·)$, $\chi_{1107}(860,·)$, $\chi_{1107}(862,·)$, $\chi_{1107}(737,·)$, $\chi_{1107}(739,·)$, $\chi_{1107}(614,·)$, $\chi_{1107}(616,·)$, $\chi_{1107}(491,·)$, $\chi_{1107}(493,·)$, $\chi_{1107}(368,·)$, $\chi_{1107}(370,·)$, $\chi_{1107}(245,·)$, $\chi_{1107}(247,·)$, $\chi_{1107}(122,·)$, $\chi_{1107}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{21571} a^{9} + \frac{90}{21571} a^{7} + \frac{2700}{21571} a^{5} + \frac{8429}{21571} a^{3} + \frac{3716}{21571} a - \frac{7468}{21571}$, $\frac{1}{21571} a^{10} + \frac{90}{21571} a^{8} + \frac{2700}{21571} a^{6} + \frac{8429}{21571} a^{4} + \frac{3716}{21571} a^{2} - \frac{7468}{21571} a$, $\frac{1}{21571} a^{11} - \frac{5400}{21571} a^{7} + \frac{2710}{21571} a^{5} + \frac{91}{21571} a^{3} - \frac{7468}{21571} a^{2} + \frac{10696}{21571} a + \frac{3419}{21571}$, $\frac{1}{21571} a^{12} - \frac{5400}{21571} a^{8} + \frac{2710}{21571} a^{6} + \frac{91}{21571} a^{4} - \frac{7468}{21571} a^{3} + \frac{10696}{21571} a^{2} + \frac{3419}{21571} a$, $\frac{1}{21571} a^{13} - \frac{7423}{21571} a^{7} - \frac{1905}{21571} a^{5} - \frac{7468}{21571} a^{4} - \frac{9085}{21571} a^{3} + \frac{3419}{21571} a^{2} + \frac{5370}{21571} a + \frac{10570}{21571}$, $\frac{1}{346810549848851} a^{14} + \frac{3715540207}{346810549848851} a^{13} + \frac{140}{346810549848851} a^{12} + \frac{691336480}{346810549848851} a^{11} + \frac{700}{31528231804441} a^{10} + \frac{2411564638}{346810549848851} a^{9} + \frac{210000}{346810549848851} a^{8} - \frac{29605349210133}{346810549848851} a^{7} + \frac{2940000}{346810549848851} a^{6} + \frac{157769757711691}{346810549848851} a^{5} - \frac{60451868000560}{346810549848851} a^{4} - \frac{55709098584475}{346810549848851} a^{3} + \frac{143525049162287}{346810549848851} a^{2} - \frac{158236190948702}{346810549848851} a + \frac{133219259536766}{346810549848851}$, $\frac{1}{346810549848851} a^{15} + \frac{150}{346810549848851} a^{13} - \frac{454558428}{31528231804441} a^{12} + \frac{9000}{346810549848851} a^{11} - \frac{5144826763}{346810549848851} a^{10} + \frac{25000}{31528231804441} a^{9} + \frac{26537736214530}{346810549848851} a^{8} + \frac{4500000}{346810549848851} a^{7} - \frac{120855528559485}{346810549848851} a^{6} + \frac{37800000}{346810549848851} a^{5} + \frac{146885353696166}{346810549848851} a^{4} + \frac{17572989241333}{346810549848851} a^{3} + \frac{10386590821526}{346810549848851} a^{2} - \frac{100404647357845}{346810549848851} a + \frac{102720221613909}{346810549848851}$, $\frac{1}{346810549848851} a^{16} + \frac{385865077}{346810549848851} a^{13} - \frac{12000}{346810549848851} a^{12} + \frac{99948892}{9373258104023} a^{11} - \frac{80000}{31528231804441} a^{10} + \frac{1620398162}{346810549848851} a^{9} - \frac{27000000}{346810549848851} a^{8} - \frac{46046030322852}{346810549848851} a^{7} - \frac{403200000}{346810549848851} a^{6} + \frac{69700005791985}{346810549848851} a^{5} + \frac{27634645421639}{346810549848851} a^{4} + \frac{2461211348719}{9373258104023} a^{3} - \frac{83884194045777}{346810549848851} a^{2} + \frac{161813377306903}{346810549848851} a + \frac{91366169477123}{346810549848851}$, $\frac{1}{346810549848851} a^{17} - \frac{13600}{346810549848851} a^{13} - \frac{2090112733}{346810549848851} a^{12} - \frac{1088000}{346810549848851} a^{11} + \frac{4820796247}{346810549848851} a^{10} - \frac{3400000}{31528231804441} a^{9} - \frac{119842763136402}{346810549848851} a^{8} - \frac{652800000}{346810549848851} a^{7} + \frac{160532837400995}{346810549848851} a^{6} - \frac{5712000000}{346810549848851} a^{5} - \frac{5288553708607}{31528231804441} a^{4} - \frac{91921491256596}{346810549848851} a^{3} + \frac{96538398436129}{346810549848851} a^{2} - \frac{4864915813517}{31528231804441} a - \frac{135139615329443}{346810549848851}$

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

Class group and class number

$C_{19}\times C_{42446}$, which has order $806474$ (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:  \( 40934.0329443 \) (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

\(\Q(\sqrt{-123}) \), \(\Q(\zeta_{9})^+\), 6.0.1356572043.1, \(\Q(\zeta_{27})^+\)

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 $18$ R $18$ $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ $18$

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
41Data not computed