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Properties

Label 18.0.21397106810...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 5^{9}\cdot 19^{17}$
Root discriminant $374.89$
Ramified primes $2, 3, 5, 19$
Class number $2473694832$ (GRH)
Class group $[2, 2, 618423708]$ (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![86016673828125, 0, 36516983203125, 0, 6230705625000, 0, 553885031250, 0, 27846281250, 0, 809364375, 0, 13494750, 0, 123975, 0, 570, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 570*x^16 + 123975*x^14 + 13494750*x^12 + 809364375*x^10 + 27846281250*x^8 + 553885031250*x^6 + 6230705625000*x^4 + 36516983203125*x^2 + 86016673828125)
 
gp: K = bnfinit(x^18 + 570*x^16 + 123975*x^14 + 13494750*x^12 + 809364375*x^10 + 27846281250*x^8 + 553885031250*x^6 + 6230705625000*x^4 + 36516983203125*x^2 + 86016673828125, 1)
 

Normalized defining polynomial

\( x^{18} + 570 x^{16} + 123975 x^{14} + 13494750 x^{12} + 809364375 x^{10} + 27846281250 x^{8} + 553885031250 x^{6} + 6230705625000 x^{4} + 36516983203125 x^{2} + 86016673828125 \)

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:  \(-21397106810613552737877882027518870016000000000=-\,2^{18}\cdot 3^{27}\cdot 5^{9}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $374.89$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3420=2^{2}\cdot 3^{2}\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{3420}(1,·)$, $\chi_{3420}(299,·)$, $\chi_{3420}(2821,·)$, $\chi_{3420}(839,·)$, $\chi_{3420}(2221,·)$, $\chi_{3420}(2581,·)$, $\chi_{3420}(599,·)$, $\chi_{3420}(2159,·)$, $\chi_{3420}(481,·)$, $\chi_{3420}(3419,·)$, $\chi_{3420}(3121,·)$, $\chi_{3420}(3241,·)$, $\chi_{3420}(2219,·)$, $\chi_{3420}(1261,·)$, $\chi_{3420}(1199,·)$, $\chi_{3420}(1201,·)$, $\chi_{3420}(179,·)$, $\chi_{3420}(2939,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{2625} a^{6} - \frac{2}{7}$, $\frac{1}{2625} a^{7} - \frac{2}{7} a$, $\frac{1}{13125} a^{8} - \frac{2}{35} a^{2}$, $\frac{1}{13125} a^{9} - \frac{2}{35} a^{3}$, $\frac{1}{65625} a^{10} - \frac{2}{175} a^{4}$, $\frac{1}{65625} a^{11} - \frac{2}{175} a^{5}$, $\frac{1}{254953125} a^{12} - \frac{1}{2428125} a^{10} + \frac{2}{69375} a^{8} - \frac{1}{27195} a^{6} + \frac{58}{6475} a^{4} - \frac{12}{185} a^{2} - \frac{591}{1813}$, $\frac{1}{74701265625} a^{13} + \frac{4513}{711440625} a^{11} - \frac{604}{28457625} a^{9} - \frac{30328}{199203375} a^{7} - \frac{3013}{1897175} a^{5} + \frac{6843}{75887} a^{3} + \frac{87210}{531209} a$, $\frac{1}{2614544296875} a^{14} - \frac{149}{104581771875} a^{12} + \frac{27452}{4980084375} a^{10} + \frac{11378}{464807875} a^{8} + \frac{48067}{1394423625} a^{6} + \frac{2448}{2656045} a^{4} - \frac{1118778}{18592315} a^{2} + \frac{3337}{12691}$, $\frac{1}{2614544296875} a^{15} - \frac{1}{174302953125} a^{13} + \frac{7414}{996016875} a^{11} - \frac{2977}{188435625} a^{9} - \frac{48177}{464807875} a^{7} + \frac{204978}{13280225} a^{5} + \frac{763753}{18592315} a^{3} - \frac{1244773}{3718463} a$, $\frac{1}{7595251182421875} a^{16} + \frac{13}{168783359609375} a^{14} + \frac{73814}{101270015765625} a^{12} + \frac{1658189}{6751334384375} a^{10} - \frac{108891488}{4050800630625} a^{8} + \frac{9379262}{270053375375} a^{6} - \frac{742424364}{54010675075} a^{4} - \frac{232910378}{10802135015} a^{2} - \frac{1504547}{7373471}$, $\frac{1}{7595251182421875} a^{17} + \frac{13}{168783359609375} a^{15} + \frac{608}{101270015765625} a^{13} - \frac{143064233}{20254003153125} a^{11} - \frac{95653403}{4050800630625} a^{9} - \frac{101200948}{810160126125} a^{7} - \frac{209715998}{10802135015} a^{5} - \frac{365006538}{10802135015} a^{3} + \frac{467389834}{2160427003} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{618423708}$, which has order $2473694832$ (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:  \( 22027035.20428972 \) (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

\(\Q(\sqrt{-285}) \), 3.3.361.1, 6.0.534837384000.9, 9.9.9025761726072081.2

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 ${\href{/LocalNumberField/7.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ $18$ $18$

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
2Data not computed
3Data not computed
5Data not computed
19Data not computed