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Properties

Label 18.0.71812000225...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{45}\cdot 5^{9}\cdot 7^{15}$
Root discriminant $352.83$
Ramified primes $2, 3, 5, 7$
Class number $686340744$ (GRH)
Class group $[2, 2, 171585186]$ (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![209676837890625, 0, 182401906640625, 0, 34743220312500, 0, 2547836156250, 0, 93593981250, 0, 1931304375, 0, 23409750, 0, 165375, 0, 630, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 630*x^16 + 165375*x^14 + 23409750*x^12 + 1931304375*x^10 + 93593981250*x^8 + 2547836156250*x^6 + 34743220312500*x^4 + 182401906640625*x^2 + 209676837890625)
 
gp: K = bnfinit(x^18 + 630*x^16 + 165375*x^14 + 23409750*x^12 + 1931304375*x^10 + 93593981250*x^8 + 2547836156250*x^6 + 34743220312500*x^4 + 182401906640625*x^2 + 209676837890625, 1)
 

Normalized defining polynomial

\( x^{18} + 630 x^{16} + 165375 x^{14} + 23409750 x^{12} + 1931304375 x^{10} + 93593981250 x^{8} + 2547836156250 x^{6} + 34743220312500 x^{4} + 182401906640625 x^{2} + 209676837890625 \)

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:  \(-7181200022505608756129304837174962688000000000=-\,2^{18}\cdot 3^{45}\cdot 5^{9}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $352.83$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3780=2^{2}\cdot 3^{3}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{3780}(1,·)$, $\chi_{3780}(3659,·)$, $\chi_{3780}(3779,·)$, $\chi_{3780}(2519,·)$, $\chi_{3780}(1739,·)$, $\chi_{3780}(781,·)$, $\chi_{3780}(2641,·)$, $\chi_{3780}(121,·)$, $\chi_{3780}(2521,·)$, $\chi_{3780}(1381,·)$, $\chi_{3780}(3301,·)$, $\chi_{3780}(1259,·)$, $\chi_{3780}(1261,·)$, $\chi_{3780}(1139,·)$, $\chi_{3780}(479,·)$, $\chi_{3780}(2999,·)$, $\chi_{3780}(2041,·)$, $\chi_{3780}(2399,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{875} a^{6}$, $\frac{1}{875} a^{7}$, $\frac{1}{4375} a^{8}$, $\frac{1}{1413125} a^{9} + \frac{9}{40375} a^{7} - \frac{134}{8075} a^{5} - \frac{29}{323} a^{3} - \frac{143}{323} a$, $\frac{1}{2805053125} a^{10} + \frac{10399}{561010625} a^{8} - \frac{61016}{112202125} a^{6} + \frac{37969}{3205775} a^{4} + \frac{27958}{641155} a^{2} + \frac{117}{397}$, $\frac{1}{2805053125} a^{11} + \frac{11}{80144375} a^{9} + \frac{58084}{112202125} a^{7} + \frac{10576}{3205775} a^{5} - \frac{14124}{641155} a^{3} - \frac{24935}{128231} a$, $\frac{1}{98176859375} a^{12} + \frac{3821}{561010625} a^{8} + \frac{40597}{112202125} a^{6} - \frac{3333}{641155} a^{4} + \frac{318}{641155} a^{2} - \frac{96}{397}$, $\frac{1}{98176859375} a^{13} - \frac{149}{561010625} a^{9} + \frac{353}{843625} a^{7} + \frac{2391}{3205775} a^{5} + \frac{63044}{641155} a^{3} + \frac{23778}{128231} a$, $\frac{1}{490884296875} a^{14} + \frac{57628}{561010625} a^{8} + \frac{1566}{5905375} a^{6} - \frac{526}{33745} a^{4} - \frac{42103}{641155} a^{2} - \frac{35}{397}$, $\frac{1}{490884296875} a^{15} + \frac{9}{80144375} a^{9} - \frac{6373}{112202125} a^{7} - \frac{6024}{641155} a^{5} - \frac{30193}{641155} a^{3} + \frac{13706}{128231} a$, $\frac{1}{2454421484375} a^{16} - \frac{4071}{112202125} a^{8} + \frac{244}{641155} a^{6} + \frac{14149}{3205775} a^{4} + \frac{47586}{641155} a^{2} + \frac{172}{397}$, $\frac{1}{2454421484375} a^{17} - \frac{108}{561010625} a^{9} + \frac{35951}{112202125} a^{7} - \frac{6098}{3205775} a^{5} + \frac{61084}{641155} a^{3} - \frac{1099}{7543} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{171585186}$, which has order $686340744$ (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:  \( 10392888.21418944 \) (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

\(\Q(\sqrt{-105}) \), \(\Q(\zeta_{9})^+\), 6.0.54010152000.17, 9.9.3691950281939241.2

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 R R R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ $18$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ $18$

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
2Data not computed
3Data not computed
5Data not computed
7Data not computed