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Properties

Label 18.0.59271764018...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 5^{9}\cdot 19^{15}$
Root discriminant $270.29$
Ramified primes $2, 3, 5, 19$
Class number $289816576$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 4, 152, 7448]$ (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![361705078125, 0, 434046093750, 0, 188086640625, 0, 38175750000, 0, 4020637500, 0, 227430000, 0, 6783000, 0, 98325, 0, 570, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 570*x^16 + 98325*x^14 + 6783000*x^12 + 227430000*x^10 + 4020637500*x^8 + 38175750000*x^6 + 188086640625*x^4 + 434046093750*x^2 + 361705078125)
 
gp: K = bnfinit(x^18 + 570*x^16 + 98325*x^14 + 6783000*x^12 + 227430000*x^10 + 4020637500*x^8 + 38175750000*x^6 + 188086640625*x^4 + 434046093750*x^2 + 361705078125, 1)
 

Normalized defining polynomial

\( x^{18} + 570 x^{16} + 98325 x^{14} + 6783000 x^{12} + 227430000 x^{10} + 4020637500 x^{8} + 38175750000 x^{6} + 188086640625 x^{4} + 434046093750 x^{2} + 361705078125 \)

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:  \(-59271764018320090686642332486201856000000000=-\,2^{18}\cdot 3^{27}\cdot 5^{9}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $270.29$
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$$\chi_{3420}(1,·)$, $\chi_{3420}(179,·)$, $\chi_{3420}(961,·)$, $\chi_{3420}(3419,·)$, $\chi_{3420}(2401,·)$, $\chi_{3420}(3299,·)$, $\chi_{3420}(1319,·)$, $\chi_{3420}(3241,·)$, $\chi_{3420}(2279,·)$, $\chi_{3420}(1261,·)$, $\chi_{3420}(2159,·)$, $\chi_{3420}(1139,·)$, $\chi_{3420}(2101,·)$, $\chi_{3420}(2281,·)$, $\chi_{3420}(121,·)$, $\chi_{3420}(1019,·)$, $\chi_{3420}(2459,·)$, $\chi_{3420}(1141,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{7125} a^{6}$, $\frac{1}{7125} a^{7}$, $\frac{1}{35625} a^{8}$, $\frac{1}{35625} a^{9}$, $\frac{1}{178125} a^{10}$, $\frac{1}{178125} a^{11}$, $\frac{1}{355359375} a^{12} - \frac{2}{1246875} a^{10} + \frac{1}{83125} a^{8} - \frac{1}{49875} a^{6} + \frac{3}{35} a^{2} + \frac{2}{7}$, $\frac{1}{355359375} a^{13} - \frac{2}{1246875} a^{11} + \frac{1}{83125} a^{9} - \frac{1}{49875} a^{7} + \frac{3}{35} a^{3} + \frac{2}{7} a$, $\frac{1}{35535937500} a^{14} + \frac{2}{1776796875} a^{12} - \frac{13}{6234375} a^{10} + \frac{2}{1246875} a^{8} + \frac{1}{249375} a^{6} - \frac{8}{875} a^{4} - \frac{6}{175} a^{2} + \frac{13}{140}$, $\frac{1}{35535937500} a^{15} + \frac{2}{1776796875} a^{13} - \frac{13}{6234375} a^{11} + \frac{2}{1246875} a^{9} + \frac{1}{249375} a^{7} - \frac{8}{875} a^{5} - \frac{6}{175} a^{3} + \frac{13}{140} a$, $\frac{1}{2498176406250000} a^{16} - \frac{61}{26296593750000} a^{14} + \frac{193}{262965937500} a^{12} + \frac{81623}{87655312500} a^{10} - \frac{3757}{922687500} a^{8} - \frac{4271}{92268750} a^{6} - \frac{14299}{1230250} a^{4} - \frac{25441}{518000} a^{2} + \frac{48411}{103600}$, $\frac{1}{2498176406250000} a^{17} - \frac{61}{26296593750000} a^{15} + \frac{193}{262965937500} a^{13} + \frac{81623}{87655312500} a^{11} - \frac{3757}{922687500} a^{9} - \frac{4271}{92268750} a^{7} - \frac{14299}{1230250} a^{5} - \frac{25441}{518000} a^{3} + \frac{48411}{103600} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{152}\times C_{7448}$, which has order $289816576$ (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:  \( 1472619.082400847 \) (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

\(\Q(\sqrt{-285}) \), \(\Q(\zeta_{9})^+\), 3.3.29241.1, 3.3.29241.2, 3.3.361.1, 6.0.1080045576000.11, 6.0.389896452936000.2, 6.0.389896452936000.1, 6.0.534837384000.9, 9.9.25002110044521.1

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 R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.6.5.2$x^{6} - 19$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.2$x^{6} - 19$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.2$x^{6} - 19$$6$$1$$5$$C_6$$[\ ]_{6}$