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Properties

Label 18.0.80308148531...3984.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 7^{12}\cdot 19^{12}$
Root discriminant $52.11$
Ramified primes $2, 7, 19$
Class number $196$ (GRH)
Class group $[14, 14]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3136, 0, 32144, 0, 116984, 0, 187993, 0, 141146, 0, 52581, 0, 10198, 0, 1038, 0, 52, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 52*x^16 + 1038*x^14 + 10198*x^12 + 52581*x^10 + 141146*x^8 + 187993*x^6 + 116984*x^4 + 32144*x^2 + 3136)
 
gp: K = bnfinit(x^18 + 52*x^16 + 1038*x^14 + 10198*x^12 + 52581*x^10 + 141146*x^8 + 187993*x^6 + 116984*x^4 + 32144*x^2 + 3136, 1)
 

Normalized defining polynomial

\( x^{18} + 52 x^{16} + 1038 x^{14} + 10198 x^{12} + 52581 x^{10} + 141146 x^{8} + 187993 x^{6} + 116984 x^{4} + 32144 x^{2} + 3136 \)

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:  \(-8030814853150226545771901353984=-\,2^{18}\cdot 7^{12}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.11$
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:  $2, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(532=2^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{532}(1,·)$, $\chi_{532}(387,·)$, $\chi_{532}(197,·)$, $\chi_{532}(457,·)$, $\chi_{532}(11,·)$, $\chi_{532}(463,·)$, $\chi_{532}(277,·)$, $\chi_{532}(121,·)$, $\chi_{532}(267,·)$, $\chi_{532}(163,·)$, $\chi_{532}(39,·)$, $\chi_{532}(235,·)$, $\chi_{532}(429,·)$, $\chi_{532}(239,·)$, $\chi_{532}(305,·)$, $\chi_{532}(501,·)$, $\chi_{532}(505,·)$, $\chi_{532}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{12} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} + \frac{5}{12} a^{2} + \frac{1}{3}$, $\frac{1}{168} a^{9} - \frac{5}{28} a^{7} - \frac{5}{56} a^{5} - \frac{55}{168} a^{3} - \frac{1}{3} a$, $\frac{1}{168} a^{10} - \frac{1}{84} a^{8} - \frac{5}{56} a^{6} + \frac{29}{168} a^{4} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{168} a^{11} + \frac{3}{56} a^{7} - \frac{1}{168} a^{5} + \frac{29}{84} a^{3} - \frac{1}{2} a$, $\frac{1}{168} a^{12} - \frac{5}{168} a^{8} + \frac{83}{168} a^{6} - \frac{17}{42} a^{4} + \frac{1}{12} a^{2} - \frac{1}{3}$, $\frac{1}{168} a^{13} + \frac{17}{168} a^{7} + \frac{25}{168} a^{5} - \frac{3}{56} a^{3} - \frac{1}{2} a$, $\frac{1}{168} a^{14} + \frac{1}{56} a^{8} - \frac{59}{168} a^{6} + \frac{11}{56} a^{4} + \frac{1}{12} a^{2} - \frac{1}{3}$, $\frac{1}{168} a^{15} + \frac{31}{168} a^{7} + \frac{13}{28} a^{5} + \frac{11}{168} a^{3} - \frac{1}{3} a$, $\frac{1}{3841851888} a^{16} + \frac{834667}{320154324} a^{14} + \frac{372635}{213436216} a^{12} - \frac{94429}{480231486} a^{10} + \frac{156694733}{3841851888} a^{8} + \frac{91803325}{320154324} a^{6} + \frac{1883269303}{3841851888} a^{4} - \frac{59616593}{137208996} a^{2} + \frac{3603274}{34302249}$, $\frac{1}{7683703776} a^{17} + \frac{834667}{640308648} a^{15} + \frac{372635}{426872432} a^{13} + \frac{1579481}{548835984} a^{11} + \frac{19485737}{7683703776} a^{9} - \frac{56509093}{1280617296} a^{7} - \frac{3765167699}{7683703776} a^{5} + \frac{360201493}{1920925944} a^{3} + \frac{1801637}{34302249} a$

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

Class group and class number

$C_{14}\times C_{14}$, which has order $196$ (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:  \( \frac{19}{5884} a^{17} + \frac{41207}{247128} a^{15} + \frac{33893}{10297} a^{13} + \frac{1307505}{41188} a^{11} + \frac{3262495}{20594} a^{9} + \frac{98746619}{247128} a^{7} + \frac{4815211}{10297} a^{5} + \frac{54126943}{247128} a^{3} + \frac{285539}{8826} a \) (order $4$)
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:  \( 833965.243856 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{7})^+\), 3.3.361.1, 3.3.17689.2, 3.3.17689.1, 6.0.153664.1, 6.0.8340544.1, 6.0.20025646144.1, 6.0.20025646144.2, 9.9.5534900853769.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19Data not computed