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Properties

Label 18.0.21972413945...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{28}\cdot 5^{6}\cdot 107^{6}$
Root discriminant $71.17$
Ramified primes $2, 3, 5, 107$
Class number $3744$ (GRH)
Class group $[2, 2, 2, 2, 234]$ (GRH)
Galois group 18T367

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![140625, 0, 759375, 0, 1395000, 0, 1174125, 0, 518625, 0, 128115, 0, 18129, 0, 1446, 0, 60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 60*x^16 + 1446*x^14 + 18129*x^12 + 128115*x^10 + 518625*x^8 + 1174125*x^6 + 1395000*x^4 + 759375*x^2 + 140625)
 
gp: K = bnfinit(x^18 + 60*x^16 + 1446*x^14 + 18129*x^12 + 128115*x^10 + 518625*x^8 + 1174125*x^6 + 1395000*x^4 + 759375*x^2 + 140625, 1)
 

Normalized defining polynomial

\( x^{18} + 60 x^{16} + 1446 x^{14} + 18129 x^{12} + 128115 x^{10} + 518625 x^{8} + 1174125 x^{6} + 1395000 x^{4} + 759375 x^{2} + 140625 \)

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:  \(-2197241394567050880996664896000000=-\,2^{12}\cdot 3^{28}\cdot 5^{6}\cdot 107^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.17$
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:  $2, 3, 5, 107$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{10} - \frac{4}{25} a^{6} - \frac{6}{25} a^{4}$, $\frac{1}{25} a^{11} - \frac{4}{25} a^{7} - \frac{6}{25} a^{5}$, $\frac{1}{750} a^{12} - \frac{1}{50} a^{10} + \frac{7}{250} a^{8} - \frac{107}{250} a^{6} + \frac{1}{50} a^{4} - \frac{1}{10} a^{2} - \frac{1}{2}$, $\frac{1}{750} a^{13} - \frac{1}{50} a^{11} + \frac{7}{250} a^{9} - \frac{107}{250} a^{7} + \frac{1}{50} a^{5} - \frac{1}{10} a^{3} - \frac{1}{2} a$, $\frac{1}{3750} a^{14} - \frac{9}{625} a^{10} - \frac{1}{625} a^{8} + \frac{9}{25} a^{6} + \frac{2}{5} a^{4} - \frac{1}{5} a^{2} - \frac{1}{2}$, $\frac{1}{7500} a^{15} - \frac{1}{7500} a^{14} - \frac{1}{1500} a^{13} - \frac{43}{2500} a^{11} - \frac{8}{625} a^{10} + \frac{213}{2500} a^{9} - \frac{62}{625} a^{8} - \frac{13}{500} a^{7} - \frac{1}{10} a^{6} - \frac{9}{100} a^{5} + \frac{8}{25} a^{4} + \frac{7}{20} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{368386612500} a^{16} + \frac{5279}{491182150} a^{14} - \frac{1}{1500} a^{13} + \frac{81464921}{368386612500} a^{12} + \frac{1}{100} a^{11} - \frac{704705377}{122795537500} a^{10} - \frac{7}{500} a^{9} - \frac{158126079}{4911821500} a^{8} - \frac{143}{500} a^{7} + \frac{2052770367}{4911821500} a^{6} - \frac{1}{100} a^{5} + \frac{194608789}{982364300} a^{4} - \frac{9}{20} a^{3} + \frac{19498178}{49118215} a^{2} - \frac{1}{4} a - \frac{15708353}{39294572}$, $\frac{1}{368386612500} a^{17} + \frac{5279}{491182150} a^{15} - \frac{1}{7500} a^{14} + \frac{81464921}{368386612500} a^{13} - \frac{1}{1500} a^{12} - \frac{704705377}{122795537500} a^{11} - \frac{7}{2500} a^{10} - \frac{158126079}{4911821500} a^{9} + \frac{217}{2500} a^{8} + \frac{2052770367}{4911821500} a^{7} - \frac{193}{500} a^{6} + \frac{194608789}{982364300} a^{5} - \frac{49}{100} a^{4} + \frac{19498178}{49118215} a^{3} - \frac{9}{20} a^{2} - \frac{15708353}{39294572} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{234}$, which has order $3744$ (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:  \( 664139.073753 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T367:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 48 conjugacy class representatives for t18n367
Character table for t18n367 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.321.1, 9.9.1953114230889.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 sibling: data not computed

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.12.19.43$x^{12} + 3 x^{10} - 3 x^{9} - 3 x^{8} - 3 x^{6} + 3 x^{3} + 3$$12$$1$$19$$D_4 \times C_3$$[2]_{4}^{2}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$107$107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$