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Properties

Label 18.0.25268488870...0000.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{31}\cdot 5^{9}\cdot 19^{14}$
Root discriminant $292.96$
Ramified primes $2, 3, 5, 19$
Class number $187146504$ (GRH)
Class group $[6, 6, 5198514]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4958060544, -7430247936, 6631652736, -3807067392, 1700179632, -660915288, 256504212, -86162778, 18161604, 174591, -1217997, 183762, 69750, -27972, 2088, 552, -90, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 90*x^16 + 552*x^15 + 2088*x^14 - 27972*x^13 + 69750*x^12 + 183762*x^11 - 1217997*x^10 + 174591*x^9 + 18161604*x^8 - 86162778*x^7 + 256504212*x^6 - 660915288*x^5 + 1700179632*x^4 - 3807067392*x^3 + 6631652736*x^2 - 7430247936*x + 4958060544)
 
gp: K = bnfinit(x^18 - 3*x^17 - 90*x^16 + 552*x^15 + 2088*x^14 - 27972*x^13 + 69750*x^12 + 183762*x^11 - 1217997*x^10 + 174591*x^9 + 18161604*x^8 - 86162778*x^7 + 256504212*x^6 - 660915288*x^5 + 1700179632*x^4 - 3807067392*x^3 + 6631652736*x^2 - 7430247936*x + 4958060544, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 90 x^{16} + 552 x^{15} + 2088 x^{14} - 27972 x^{13} + 69750 x^{12} + 183762 x^{11} - 1217997 x^{10} + 174591 x^{9} + 18161604 x^{8} - 86162778 x^{7} + 256504212 x^{6} - 660915288 x^{5} + 1700179632 x^{4} - 3807067392 x^{3} + 6631652736 x^{2} - 7430247936 x + 4958060544 \)

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:  \(-252684888709680386611475206914860544000000000=-\,2^{18}\cdot 3^{31}\cdot 5^{9}\cdot 19^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $292.96$
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 a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{6} - \frac{1}{12} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{12} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{24} a^{8} - \frac{1}{12} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{216} a^{9} + \frac{1}{72} a^{8} + \frac{1}{36} a^{6} - \frac{1}{24} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{432} a^{10} - \frac{1}{432} a^{9} + \frac{1}{72} a^{8} + \frac{1}{72} a^{7} + \frac{1}{144} a^{6} + \frac{1}{48} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{432} a^{11} - \frac{1}{432} a^{9} - \frac{1}{72} a^{8} + \frac{1}{48} a^{7} + \frac{1}{36} a^{6} + \frac{1}{48} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{12} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{1728} a^{12} - \frac{1}{864} a^{11} - \frac{1}{1728} a^{10} - \frac{1}{576} a^{8} - \frac{7}{288} a^{7} + \frac{5}{192} a^{6} - \frac{1}{48} a^{5} - \frac{3}{32} a^{4} - \frac{1}{12} a^{3} + \frac{7}{24} a^{2} - \frac{1}{6} a$, $\frac{1}{1728} a^{13} - \frac{1}{1728} a^{11} - \frac{1}{864} a^{10} + \frac{1}{1728} a^{9} + \frac{1}{72} a^{8} - \frac{1}{576} a^{7} + \frac{1}{288} a^{6} + \frac{1}{96} a^{5} - \frac{7}{48} a^{4} + \frac{1}{4} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{10368} a^{14} - \frac{1}{3456} a^{13} - \frac{1}{3456} a^{12} + \frac{1}{1152} a^{11} + \frac{1}{1152} a^{10} - \frac{5}{3456} a^{9} - \frac{7}{384} a^{8} + \frac{5}{384} a^{7} - \frac{5}{576} a^{6} + \frac{13}{576} a^{5} + \frac{5}{24} a^{4} - \frac{7}{48} a^{3} + \frac{1}{12} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{10368} a^{15} - \frac{1}{864} a^{10} - \frac{1}{1728} a^{9} + \frac{1}{72} a^{8} + \frac{7}{1152} a^{7} + \frac{1}{288} a^{6} - \frac{11}{192} a^{5} - \frac{1}{16} a^{4} - \frac{23}{48} a^{3} - \frac{1}{4} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{41472} a^{16} - \frac{1}{20736} a^{14} - \frac{1}{6912} a^{13} + \frac{1}{6912} a^{12} + \frac{5}{6912} a^{11} + \frac{11}{6912} a^{9} - \frac{79}{4608} a^{8} - \frac{83}{2304} a^{7} - \frac{11}{2304} a^{6} - \frac{37}{1152} a^{5} + \frac{43}{192} a^{4} - \frac{23}{96} a^{3} + \frac{7}{24} a^{2} - \frac{1}{4} a + \frac{1}{3}$, $\frac{1}{62968782306044578672335133879595263066576896} a^{17} + \frac{71237496591001281943450281240283543709}{7871097788255572334041891734949407883322112} a^{16} - \frac{471319183502079950816518519504625959573}{31484391153022289336167566939797631533288448} a^{15} + \frac{1271964586775357193975225722424468916541}{31484391153022289336167566939797631533288448} a^{14} - \frac{906720109869180500235023584568163054535}{10494797051007429778722522313265877177762816} a^{13} - \frac{2673472111651537768000168903218937586395}{10494797051007429778722522313265877177762816} a^{12} + \frac{787658540791205286538732587880618119763}{1311849631375928722340315289158234647220352} a^{11} - \frac{5163885972977999776206177988963526507581}{10494797051007429778722522313265877177762816} a^{10} + \frac{4196000280747058304796172404643357371697}{6996531367338286519148348208843918118508544} a^{9} + \frac{135253097637928947404153187879896966113}{6514461235882948341851348425366776646656} a^{8} + \frac{130927118208568460055250563998184405074273}{3498265683669143259574174104421959059254272} a^{7} + \frac{28816433334163215847490532639678054974327}{1749132841834571629787087052210979529627136} a^{6} + \frac{31092444054925293012971686813181439993727}{874566420917285814893543526105489764813568} a^{5} - \frac{4616061906485431332622430987509904879083}{145761070152880969148923921017581627468928} a^{4} - \frac{1052396571240633266004828423038369667409}{4555033442277530285903872531799425858404} a^{3} - \frac{4808617666036859688079545869275562762185}{18220133769110121143615490127197703433616} a^{2} - \frac{813740560195445706684642908953202592953}{4555033442277530285903872531799425858404} a + \frac{92968897857747878354165256371655792145}{379586120189794190491989377649952154867}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{5198514}$, which has order $187146504$ (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:  \( 15773424688.63964 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.3.29241.1, 3.3.4104.1, 6.0.320638530375.8, 6.0.6316056000.5, 9.9.6566954215853707776.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_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$19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
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}$