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Properties

Label 18.0.76198610101...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{26}\cdot 3^{31}\cdot 5^{6}\cdot 7^{6}$
Root discriminant $59.05$
Ramified primes $2, 3, 5, 7$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![140608, 32448, -73008, 29136, -65244, -13680, 11553, -9060, 19710, -1024, 6267, 0, 660, -48, -45, -12, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 - 12*x^15 - 45*x^14 - 48*x^13 + 660*x^12 + 6267*x^10 - 1024*x^9 + 19710*x^8 - 9060*x^7 + 11553*x^6 - 13680*x^5 - 65244*x^4 + 29136*x^3 - 73008*x^2 + 32448*x + 140608)
 
gp: K = bnfinit(x^18 - 6*x^16 - 12*x^15 - 45*x^14 - 48*x^13 + 660*x^12 + 6267*x^10 - 1024*x^9 + 19710*x^8 - 9060*x^7 + 11553*x^6 - 13680*x^5 - 65244*x^4 + 29136*x^3 - 73008*x^2 + 32448*x + 140608, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{16} - 12 x^{15} - 45 x^{14} - 48 x^{13} + 660 x^{12} + 6267 x^{10} - 1024 x^{9} + 19710 x^{8} - 9060 x^{7} + 11553 x^{6} - 13680 x^{5} - 65244 x^{4} + 29136 x^{3} - 73008 x^{2} + 32448 x + 140608 \)

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:  \(-76198610101243824678371328000000=-\,2^{26}\cdot 3^{31}\cdot 5^{6}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.05$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{12} a^{10} + \frac{1}{4} a^{2} + \frac{1}{3} a$, $\frac{1}{36} a^{11} - \frac{1}{36} a^{10} + \frac{1}{36} a^{9} - \frac{1}{12} a^{8} + \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{1}{12} a^{5} - \frac{1}{12} a^{4} - \frac{1}{6} a^{3} - \frac{7}{18} a^{2} + \frac{7}{18} a + \frac{1}{9}$, $\frac{1}{72} a^{12} - \frac{1}{72} a^{11} - \frac{1}{36} a^{10} - \frac{1}{12} a^{8} + \frac{1}{12} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{5}{24} a^{4} + \frac{31}{72} a^{3} - \frac{11}{36} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{72} a^{13} - \frac{1}{72} a^{11} + \frac{1}{36} a^{10} + \frac{1}{36} a^{9} - \frac{1}{12} a^{8} + \frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{24} a^{5} + \frac{5}{36} a^{4} + \frac{5}{24} a^{3} + \frac{4}{9} a^{2} + \frac{5}{18} a + \frac{1}{9}$, $\frac{1}{72} a^{14} - \frac{1}{72} a^{11} + \frac{1}{36} a^{10} - \frac{1}{36} a^{9} + \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{5}{24} a^{6} + \frac{5}{36} a^{5} + \frac{1}{12} a^{4} - \frac{5}{24} a^{3} + \frac{13}{36} a^{2} - \frac{7}{18} a - \frac{1}{9}$, $\frac{1}{144} a^{15} - \frac{1}{144} a^{11} - \frac{1}{72} a^{10} + \frac{1}{36} a^{9} + \frac{1}{12} a^{8} + \frac{5}{48} a^{7} - \frac{1}{18} a^{6} + \frac{1}{6} a^{5} + \frac{1}{12} a^{4} + \frac{11}{48} a^{3} - \frac{11}{72} a^{2} - \frac{11}{36} a + \frac{5}{18}$, $\frac{1}{18720} a^{16} - \frac{1}{360} a^{15} - \frac{1}{3120} a^{14} - \frac{1}{1560} a^{13} - \frac{97}{18720} a^{12} + \frac{1}{4680} a^{11} + \frac{5}{234} a^{10} - \frac{1}{45} a^{9} + \frac{113}{6240} a^{8} - \frac{1127}{4680} a^{7} + \frac{47}{1872} a^{6} - \frac{79}{1560} a^{5} - \frac{257}{6240} a^{4} + \frac{1091}{4680} a^{3} - \frac{233}{1170} a^{2} + \frac{98}{585} a + \frac{19}{90}$, $\frac{1}{4077167808182259957341760} a^{17} - \frac{481237670782239769}{52271382156182819965920} a^{16} + \frac{340095767891995771237}{2038583904091129978670880} a^{15} + \frac{2323926016532057211721}{509645976022782494667720} a^{14} - \frac{778655358794172504533}{453018645353584439704640} a^{13} - \frac{501401170014410254051}{679527968030376659556960} a^{12} + \frac{688769053310048046139}{67952796803037665955696} a^{11} - \frac{872929763867214595069}{39203536617137114974440} a^{10} - \frac{21268850424222214801927}{1359055936060753319113920} a^{9} - \frac{218995820627362733239069}{2038583904091129978670880} a^{8} + \frac{7897538309618655714651}{45301864535358443970464} a^{7} - \frac{87530111289773373535361}{509645976022782494667720} a^{6} - \frac{408770292211317249417871}{4077167808182259957341760} a^{5} + \frac{123268948631792116758119}{679527968030376659556960} a^{4} + \frac{19389887201345494454997}{113254661338396109926160} a^{3} - \frac{7658090954333167537547}{56627330669198054963080} a^{2} - \frac{84483202416401906909}{4900442077142139371805} a - \frac{40264775575235622}{8376824063490836533}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( -\frac{40386895}{4589362651968} a^{17} - \frac{198356707}{3824468876640} a^{16} + \frac{2137782197}{11473406629920} a^{15} + \frac{695951327}{1434175828740} a^{14} - \frac{68724049}{2549645917760} a^{13} + \frac{4510483597}{11473406629920} a^{12} - \frac{45957683597}{5736703314960} a^{11} - \frac{7352550139}{191223443832} a^{10} + \frac{900378063367}{22946813259840} a^{9} - \frac{2867539440619}{11473406629920} a^{8} + \frac{2119341620861}{3824468876640} a^{7} - \frac{135892911209}{143417582874} a^{6} + \frac{41630621281181}{22946813259840} a^{5} - \frac{2507870491521}{1274822958880} a^{4} + \frac{6025610017043}{5736703314960} a^{3} - \frac{188656004173}{2868351657480} a^{2} - \frac{462769617271}{119514652395} a + \frac{1674697350514}{358543957185} \) (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:  \( 292071104.75417435 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.1080.1, 6.0.15431472.1, 6.0.3499200.1

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

Sibling fields

Degree 12 sibling: data not computed
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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$[\ ]_{3}^{6}$
2.12.22.27$x^{12} + 2 x^{6} + 4$$6$$2$$22$$C_6\times S_3$$[3]_{3}^{6}$
3Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$