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Properties

Label 18.0.12853682015...0000.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{22}\cdot 5^{12}$
Root discriminant $28.22$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2048, -4096, 4096, -5376, 11008, -11776, 8592, -7056, 4680, -2192, 1220, -692, 282, -120, 60, -18, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 2*x^16 - 18*x^15 + 60*x^14 - 120*x^13 + 282*x^12 - 692*x^11 + 1220*x^10 - 2192*x^9 + 4680*x^8 - 7056*x^7 + 8592*x^6 - 11776*x^5 + 11008*x^4 - 5376*x^3 + 4096*x^2 - 4096*x + 2048)
 
gp: K = bnfinit(x^18 - 2*x^17 + 2*x^16 - 18*x^15 + 60*x^14 - 120*x^13 + 282*x^12 - 692*x^11 + 1220*x^10 - 2192*x^9 + 4680*x^8 - 7056*x^7 + 8592*x^6 - 11776*x^5 + 11008*x^4 - 5376*x^3 + 4096*x^2 - 4096*x + 2048, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 2 x^{16} - 18 x^{15} + 60 x^{14} - 120 x^{13} + 282 x^{12} - 692 x^{11} + 1220 x^{10} - 2192 x^{9} + 4680 x^{8} - 7056 x^{7} + 8592 x^{6} - 11776 x^{5} + 11008 x^{4} - 5376 x^{3} + 4096 x^{2} - 4096 x + 2048 \)

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:  \(-128536820158464000000000000=-\,2^{24}\cdot 3^{22}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.22$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{32} a^{12} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{13} - \frac{1}{16} a^{10} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{64} a^{14} - \frac{1}{32} a^{11} - \frac{1}{16} a^{9} + \frac{1}{32} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4}$, $\frac{1}{128} a^{15} - \frac{1}{64} a^{13} - \frac{1}{64} a^{12} + \frac{1}{64} a^{9} - \frac{1}{8} a^{8} + \frac{3}{32} a^{7} + \frac{1}{8} a^{6} + \frac{3}{16} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{15104} a^{16} + \frac{11}{7552} a^{15} + \frac{3}{7552} a^{14} - \frac{37}{7552} a^{13} - \frac{35}{3776} a^{12} + \frac{23}{944} a^{11} - \frac{299}{7552} a^{10} - \frac{49}{3776} a^{9} - \frac{7}{64} a^{8} + \frac{7}{236} a^{7} - \frac{77}{1888} a^{6} - \frac{107}{944} a^{5} - \frac{265}{944} a^{4} - \frac{22}{59} a^{3} - \frac{29}{236} a^{2} + \frac{5}{59} a - \frac{18}{59}$, $\frac{1}{96677558331516416} a^{17} + \frac{138209432613}{48338779165758208} a^{16} + \frac{160978946972491}{48338779165758208} a^{15} - \frac{24472580494149}{48338779165758208} a^{14} + \frac{205602228430991}{24169389582879104} a^{13} + \frac{18070462822421}{3021173697859888} a^{12} - \frac{512800685718339}{48338779165758208} a^{11} - \frac{905384891481207}{24169389582879104} a^{10} + \frac{1326352340289755}{24169389582879104} a^{9} - \frac{188366650023135}{1510586848929944} a^{8} - \frac{2000989262935089}{12084694791439552} a^{7} - \frac{752107397394427}{6042347395719776} a^{6} - \frac{413430758030821}{6042347395719776} a^{5} + \frac{9155392435327}{25603166931016} a^{4} + \frac{69157281742971}{1510586848929944} a^{3} - \frac{127883725946865}{377646712232486} a^{2} + \frac{1221933607058}{188823356116243} a - \frac{94067192474992}{188823356116243}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{384086823341}{409650670896256} a^{17} - \frac{44787477923}{51206333862032} a^{16} + \frac{382570375217}{409650670896256} a^{15} - \frac{396529562139}{25603166931016} a^{14} + \frac{8138547897961}{204825335448128} a^{13} - \frac{14212421500571}{204825335448128} a^{12} + \frac{37605257150095}{204825335448128} a^{11} - \frac{5648788520079}{12801583465508} a^{10} + \frac{133133438695405}{204825335448128} a^{9} - \frac{131930496688767}{102412667724064} a^{8} + \frac{293062631054767}{102412667724064} a^{7} - \frac{85698078918827}{25603166931016} a^{6} + \frac{202334526274865}{51206333862032} a^{5} - \frac{148785471030699}{25603166931016} a^{4} + \frac{82390907550621}{25603166931016} a^{3} - \frac{30182753300}{3200395866377} a^{2} + \frac{4848106204655}{3200395866377} a - \frac{4803512755001}{3200395866377} \) (order $4$)
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:  \( 5283323.889324977 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.300.1, 6.0.1440000.1, 9.3.177147000000.1

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.22.83$x^{12} + 27 x^{11} - 27 x^{10} + 3 x^{9} - 9 x^{8} - 36 x^{7} - 9 x^{6} + 27 x^{5} + 36 x^{3} + 27 x + 36$$6$$2$$22$$C_6\times S_3$$[5/2]_{2}^{6}$
$5$5.9.6.1$x^{9} - 25 x^{3} + 250$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
5.9.6.1$x^{9} - 25 x^{3} + 250$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$