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Properties

Label 18.0.47691157991...5488.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 283^{9}$
Root discriminant $26.70$
Ramified primes $2, 283$
Class number $4$
Class group $[2, 2]$
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3249, 171, -5771, -6926, 5354, 3764, 882, -196, 1687, 427, 187, 20, 262, -32, 30, 10, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 5*x^16 + 10*x^15 + 30*x^14 - 32*x^13 + 262*x^12 + 20*x^11 + 187*x^10 + 427*x^9 + 1687*x^8 - 196*x^7 + 882*x^6 + 3764*x^5 + 5354*x^4 - 6926*x^3 - 5771*x^2 + 171*x + 3249)
 
gp: K = bnfinit(x^18 - x^17 + 5*x^16 + 10*x^15 + 30*x^14 - 32*x^13 + 262*x^12 + 20*x^11 + 187*x^10 + 427*x^9 + 1687*x^8 - 196*x^7 + 882*x^6 + 3764*x^5 + 5354*x^4 - 6926*x^3 - 5771*x^2 + 171*x + 3249, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 5 x^{16} + 10 x^{15} + 30 x^{14} - 32 x^{13} + 262 x^{12} + 20 x^{11} + 187 x^{10} + 427 x^{9} + 1687 x^{8} - 196 x^{7} + 882 x^{6} + 3764 x^{5} + 5354 x^{4} - 6926 x^{3} - 5771 x^{2} + 171 x + 3249 \)

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:  \(-47691157991223085851455488=-\,2^{12}\cdot 283^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.70$
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:  $2, 283$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{6} a$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{12} a^{2} - \frac{5}{12} a + \frac{1}{4}$, $\frac{1}{12} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{10} - \frac{1}{12} a^{9} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} + \frac{1}{4} a^{3} + \frac{7}{24} a^{2} - \frac{1}{6} a + \frac{3}{8}$, $\frac{1}{120} a^{13} - \frac{1}{24} a^{11} + \frac{1}{60} a^{9} - \frac{3}{20} a^{8} + \frac{3}{40} a^{7} - \frac{1}{20} a^{6} - \frac{1}{12} a^{5} - \frac{1}{120} a^{3} - \frac{1}{20} a^{2} - \frac{41}{120} a - \frac{7}{20}$, $\frac{1}{240} a^{14} - \frac{1}{240} a^{13} - \frac{1}{48} a^{11} - \frac{1}{80} a^{10} - \frac{1}{24} a^{9} - \frac{1}{80} a^{8} - \frac{3}{16} a^{7} - \frac{19}{240} a^{6} - \frac{11}{24} a^{5} + \frac{3}{80} a^{4} - \frac{5}{48} a^{3} + \frac{19}{240} a - \frac{21}{80}$, $\frac{1}{720} a^{15} - \frac{1}{720} a^{14} - \frac{1}{360} a^{13} - \frac{1}{144} a^{12} + \frac{3}{80} a^{11} - \frac{1}{72} a^{10} + \frac{11}{240} a^{9} - \frac{23}{240} a^{8} + \frac{143}{720} a^{7} - \frac{49}{360} a^{6} + \frac{269}{720} a^{5} + \frac{7}{144} a^{4} + \frac{37}{120} a^{3} + \frac{151}{720} a^{2} - \frac{47}{240} a + \frac{1}{5}$, $\frac{1}{18000} a^{16} - \frac{1}{4500} a^{15} + \frac{7}{4500} a^{14} - \frac{17}{4500} a^{13} + \frac{4}{375} a^{12} + \frac{43}{2250} a^{11} - \frac{17}{750} a^{10} + \frac{21}{1000} a^{9} + \frac{493}{3600} a^{8} + \frac{23}{225} a^{7} + \frac{683}{4500} a^{6} - \frac{4001}{9000} a^{5} - \frac{23}{150} a^{4} - \frac{533}{1125} a^{3} - \frac{281}{1500} a^{2} + \frac{1247}{3000} a + \frac{749}{2000}$, $\frac{1}{5103770956500726000} a^{17} + \frac{8111899104241}{5103770956500726000} a^{16} + \frac{1174146061114199}{2551885478250363000} a^{15} + \frac{2251046771835221}{2551885478250363000} a^{14} + \frac{339812741807677}{318985684781295375} a^{13} + \frac{2274980755176274}{318985684781295375} a^{12} + \frac{97952313018487661}{2551885478250363000} a^{11} - \frac{50577033485982833}{1275942739125181500} a^{10} - \frac{1503266618565755}{40830167652005808} a^{9} - \frac{197311310723558777}{1020754191300145200} a^{8} + \frac{59050115420573791}{2551885478250363000} a^{7} + \frac{63990338843908268}{318985684781295375} a^{6} - \frac{5721376487678674}{12759427391251815} a^{5} + \frac{177985421072298511}{2551885478250363000} a^{4} - \frac{100365582576694633}{1275942739125181500} a^{3} - \frac{878196230233777529}{2551885478250363000} a^{2} + \frac{766844544794123327}{1701256985500242000} a - \frac{1458769281425491}{5969322756141200}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 249031.858474 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-283}) \), 3.1.283.1 x3, 6.0.22665187.1, 9.1.410511866944.1 x9

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
283Data not computed