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Properties

Label 18.0.89695807805...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{39}\cdot 5^{12}\cdot 19^{12}$
Root discriminant $357.22$
Ramified primes $2, 3, 5, 19$
Class number $8748$ (GRH)
Class group $[3, 3, 3, 3, 6, 18]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![40353607, 0, 0, -184875000, 0, 0, 313150767, 0, 0, 1705300, 0, 0, -28551, 0, 0, -60, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 60*x^15 - 28551*x^12 + 1705300*x^9 + 313150767*x^6 - 184875000*x^3 + 40353607)
 
gp: K = bnfinit(x^18 - 60*x^15 - 28551*x^12 + 1705300*x^9 + 313150767*x^6 - 184875000*x^3 + 40353607, 1)
 

Normalized defining polynomial

\( x^{18} - 60 x^{15} - 28551 x^{12} + 1705300 x^{9} + 313150767 x^{6} - 184875000 x^{3} + 40353607 \)

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:  \(-8969580780515349163336915861800987000000000000=-\,2^{12}\cdot 3^{39}\cdot 5^{12}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $357.22$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{18} a^{6} - \frac{1}{9} a^{3} + \frac{1}{18}$, $\frac{1}{18} a^{7} - \frac{1}{9} a^{4} + \frac{1}{18} a$, $\frac{1}{54} a^{8} + \frac{1}{54} a^{7} + \frac{1}{54} a^{6} - \frac{1}{27} a^{5} - \frac{1}{27} a^{4} - \frac{1}{27} a^{3} + \frac{1}{54} a^{2} + \frac{1}{54} a + \frac{1}{54}$, $\frac{1}{108} a^{9} - \frac{1}{36} a^{6} + \frac{1}{36} a^{3} - \frac{1}{108}$, $\frac{1}{108} a^{10} - \frac{1}{36} a^{7} + \frac{1}{36} a^{4} - \frac{1}{108} a$, $\frac{1}{324} a^{11} + \frac{1}{324} a^{10} + \frac{1}{324} a^{9} - \frac{1}{108} a^{8} - \frac{1}{108} a^{7} - \frac{1}{108} a^{6} + \frac{1}{108} a^{5} + \frac{1}{108} a^{4} + \frac{1}{108} a^{3} - \frac{1}{324} a^{2} - \frac{1}{324} a - \frac{1}{324}$, $\frac{1}{326916} a^{12} + \frac{779}{326916} a^{9} + \frac{2501}{108972} a^{6} - \frac{33007}{326916} a^{3} + \frac{13471}{81729}$, $\frac{1}{980748} a^{13} - \frac{1}{980748} a^{12} + \frac{779}{980748} a^{10} - \frac{779}{980748} a^{9} - \frac{3553}{326916} a^{7} + \frac{3553}{326916} a^{6} + \frac{112289}{980748} a^{4} - \frac{112289}{980748} a^{3} - \frac{36625}{490374} a + \frac{36625}{490374}$, $\frac{1}{980748} a^{14} - \frac{1}{980748} a^{12} + \frac{779}{980748} a^{11} - \frac{779}{980748} a^{9} + \frac{2501}{326916} a^{8} + \frac{1}{54} a^{7} - \frac{8555}{326916} a^{6} + \frac{75965}{980748} a^{5} - \frac{1}{27} a^{4} - \frac{39641}{980748} a^{3} - \frac{13772}{245187} a^{2} + \frac{1}{54} a + \frac{18463}{490374}$, $\frac{1}{15656247196344} a^{15} - \frac{12003005}{15656247196344} a^{12} - \frac{1228492240}{1957030899543} a^{9} + \frac{47756813059}{1957030899543} a^{6} + \frac{1004126499191}{15656247196344} a^{3} + \frac{2642949232301}{15656247196344}$, $\frac{1}{16110278365037976} a^{16} - \frac{1}{46968741589032} a^{15} - \frac{7674520445}{16110278365037976} a^{13} + \frac{59893739}{46968741589032} a^{12} - \frac{1757259871387}{8055139182518988} a^{10} + \frac{48025017881}{11742185397258} a^{9} - \frac{80524949700677}{8055139182518988} a^{7} - \frac{114406520681}{11742185397258} a^{6} + \frac{381345295989293}{16110278365037976} a^{4} - \frac{7368709266323}{46968741589032} a^{3} - \frac{295286695812985}{16110278365037976} a + \frac{5011378892185}{46968741589032}$, $\frac{1}{5525825479208025768} a^{17} - \frac{1}{46968741589032} a^{15} + \frac{912210698227}{5525825479208025768} a^{14} - \frac{35887729}{46968741589032} a^{12} - \frac{2278785279998065}{2762912739604012884} a^{11} - \frac{1}{324} a^{10} - \frac{13739471933}{23484370794516} a^{9} - \frac{806314387232885}{2762912739604012884} a^{8} - \frac{1}{108} a^{7} + \frac{64206414617}{23484370794516} a^{6} + \frac{451835352393214649}{5525825479208025768} a^{5} + \frac{1}{36} a^{4} - \frac{1162980063869}{46968741589032} a^{3} + \frac{2448708823190296007}{5525825479208025768} a^{2} - \frac{5}{324} a - \frac{4353702032249}{46968741589032}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{6}\times C_{18}$, which has order $8748$ (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:  \( -\frac{1}{73689288} a^{15} + \frac{545}{663203592} a^{12} + \frac{63995}{165800898} a^{9} - \frac{141985}{6140774} a^{6} - \frac{2804676875}{663203592} a^{3} + \frac{1158532651}{663203592} \) (order $6$)
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:  \( 4411949748464.114 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.87723.1 x3, Deg 6, Deg 6, 6.0.196830000.3, 6.0.23085974187.5

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

Sibling fields

Degree 18 siblings: 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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$19$19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$