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Properties

Label 18.0.10042857702...4368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 37^{14}\cdot 67^{9}$
Root discriminant $215.49$
Ramified primes $2, 37, 67$
Class number $45641232$ (GRH)
Class group $[2, 6, 18, 211302]$ (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![442378829803, -183041086898, 191099338775, -59468160457, 34867471171, -8598059427, 3673658915, -736480093, 253396625, -41702360, 12151554, -1636273, 414045, -44661, 9831, -794, 147, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 147*x^16 - 794*x^15 + 9831*x^14 - 44661*x^13 + 414045*x^12 - 1636273*x^11 + 12151554*x^10 - 41702360*x^9 + 253396625*x^8 - 736480093*x^7 + 3673658915*x^6 - 8598059427*x^5 + 34867471171*x^4 - 59468160457*x^3 + 191099338775*x^2 - 183041086898*x + 442378829803)
 
gp: K = bnfinit(x^18 - 7*x^17 + 147*x^16 - 794*x^15 + 9831*x^14 - 44661*x^13 + 414045*x^12 - 1636273*x^11 + 12151554*x^10 - 41702360*x^9 + 253396625*x^8 - 736480093*x^7 + 3673658915*x^6 - 8598059427*x^5 + 34867471171*x^4 - 59468160457*x^3 + 191099338775*x^2 - 183041086898*x + 442378829803, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 147 x^{16} - 794 x^{15} + 9831 x^{14} - 44661 x^{13} + 414045 x^{12} - 1636273 x^{11} + 12151554 x^{10} - 41702360 x^{9} + 253396625 x^{8} - 736480093 x^{7} + 3673658915 x^{6} - 8598059427 x^{5} + 34867471171 x^{4} - 59468160457 x^{3} + 191099338775 x^{2} - 183041086898 x + 442378829803 \)

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:  \(-1004285770264603077807571330317567473594368=-\,2^{12}\cdot 37^{14}\cdot 67^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $215.49$
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:  $2, 37, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{14021258486649930975651181137474656032795077282431695156437980900551} a^{17} + \frac{4724702649758965902930939590882584089315117244714580156160198822936}{14021258486649930975651181137474656032795077282431695156437980900551} a^{16} + \frac{6264315351770081819333478690882880680125695767888574573212981291590}{14021258486649930975651181137474656032795077282431695156437980900551} a^{15} + \frac{4062225089965735371438936718558368903265322893499382374606209750873}{14021258486649930975651181137474656032795077282431695156437980900551} a^{14} + \frac{5396551898199599424658990135794794499581063785945090351938697832032}{14021258486649930975651181137474656032795077282431695156437980900551} a^{13} - \frac{2916563921552904869436622610349600763106424747212150447165631100103}{14021258486649930975651181137474656032795077282431695156437980900551} a^{12} - \frac{2649948315035378099477567907943574632061627094736562035808978878755}{14021258486649930975651181137474656032795077282431695156437980900551} a^{11} - \frac{3322017531107984031891900378974673332886033009720674302646878276724}{14021258486649930975651181137474656032795077282431695156437980900551} a^{10} - \frac{3872960375561036451901459171800588712325935821776409640707587231193}{14021258486649930975651181137474656032795077282431695156437980900551} a^{9} + \frac{2543974249714237227122591425827337729721097623005943218869585111941}{14021258486649930975651181137474656032795077282431695156437980900551} a^{8} - \frac{81953866397086563224441018295877475278128555465961022948857728662}{452298660859675192762941327015311484928873460723603069562515512921} a^{7} + \frac{3775768871017338750919101542125140571156137087751901888660449005878}{14021258486649930975651181137474656032795077282431695156437980900551} a^{6} - \frac{2169935368773343659361956243877373413114960789244621921119820912199}{14021258486649930975651181137474656032795077282431695156437980900551} a^{5} - \frac{5532810804806034113192947724938161865639858884976475004610474634966}{14021258486649930975651181137474656032795077282431695156437980900551} a^{4} - \frac{2104883669348524295346990859062627297610311002501812819506106034503}{14021258486649930975651181137474656032795077282431695156437980900551} a^{3} + \frac{1953416034667171959707846682332628721966027454325917435509618011806}{14021258486649930975651181137474656032795077282431695156437980900551} a^{2} + \frac{3384521547468976386118094713643388879863009586854572624145543522562}{14021258486649930975651181137474656032795077282431695156437980900551} a + \frac{6653642230706155535302966234694672450223152269929426535562564982631}{14021258486649930975651181137474656032795077282431695156437980900551}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{18}\times C_{211302}$, which has order $45641232$ (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:  \( 615797.1340659427 \) (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

\(\Q(\sqrt{-67}) \), 3.3.1369.1, 3.3.148.1, 6.0.6587912752.4, 6.0.563678284843.4, 9.9.6075640136512.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
$67$67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$