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Properties

Label 18.0.19541689959...9008.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{20}\cdot 3^{9}\cdot 79^{10}$
Root discriminant $42.39$
Ramified primes $2, 3, 79$
Class number $3$ (GRH)
Class group $[3]$ (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![160003, 22997, 127507, -90118, 57104, -21324, 34702, 252, -1497, -6703, 4523, 2040, -1012, -504, 228, 54, -21, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 21*x^16 + 54*x^15 + 228*x^14 - 504*x^13 - 1012*x^12 + 2040*x^11 + 4523*x^10 - 6703*x^9 - 1497*x^8 + 252*x^7 + 34702*x^6 - 21324*x^5 + 57104*x^4 - 90118*x^3 + 127507*x^2 + 22997*x + 160003)
 
gp: K = bnfinit(x^18 - 3*x^17 - 21*x^16 + 54*x^15 + 228*x^14 - 504*x^13 - 1012*x^12 + 2040*x^11 + 4523*x^10 - 6703*x^9 - 1497*x^8 + 252*x^7 + 34702*x^6 - 21324*x^5 + 57104*x^4 - 90118*x^3 + 127507*x^2 + 22997*x + 160003, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 21 x^{16} + 54 x^{15} + 228 x^{14} - 504 x^{13} - 1012 x^{12} + 2040 x^{11} + 4523 x^{10} - 6703 x^{9} - 1497 x^{8} + 252 x^{7} + 34702 x^{6} - 21324 x^{5} + 57104 x^{4} - 90118 x^{3} + 127507 x^{2} + 22997 x + 160003 \)

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:  \(-195416899593798138941703979008=-\,2^{20}\cdot 3^{9}\cdot 79^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.39$
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:  $2, 3, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{327692} a^{15} + \frac{1467}{81923} a^{14} + \frac{383}{81923} a^{13} + \frac{1263}{327692} a^{12} + \frac{46535}{327692} a^{11} - \frac{4379}{19276} a^{10} - \frac{80905}{327692} a^{9} + \frac{7550}{81923} a^{8} - \frac{58577}{327692} a^{7} - \frac{22053}{327692} a^{6} - \frac{77839}{327692} a^{5} - \frac{25191}{81923} a^{4} - \frac{2862}{81923} a^{3} - \frac{12175}{327692} a^{2} + \frac{118733}{327692} a + \frac{755}{5372}$, $\frac{1}{327692} a^{16} - \frac{6058}{81923} a^{14} + \frac{23017}{327692} a^{13} + \frac{8321}{327692} a^{12} - \frac{10541}{327692} a^{11} - \frac{62817}{327692} a^{10} - \frac{11242}{81923} a^{9} + \frac{9195}{327692} a^{8} - \frac{41125}{327692} a^{7} + \frac{54671}{327692} a^{6} - \frac{2120}{4819} a^{5} - \frac{24255}{163846} a^{4} + \frac{151675}{327692} a^{3} - \frac{39069}{327692} a^{2} - \frac{5997}{327692} a - \frac{563}{2686}$, $\frac{1}{56819199515981322826262194220498452} a^{17} - \frac{52246132121400540921772245651}{56819199515981322826262194220498452} a^{16} + \frac{4258584829588569925263958583}{56819199515981322826262194220498452} a^{15} + \frac{3914194320460559973829829676340165}{56819199515981322826262194220498452} a^{14} - \frac{6336571939144100420888751445777061}{56819199515981322826262194220498452} a^{13} + \frac{310039504603120919165386898776809}{28409599757990661413131097110249226} a^{12} - \frac{1682446519800784647577681075112548}{14204799878995330706565548555124613} a^{11} - \frac{12425470470063403406532955905406615}{56819199515981322826262194220498452} a^{10} - \frac{2689315427665197823832213414094131}{28409599757990661413131097110249226} a^{9} + \frac{13043806552521948385499050340057}{179807593405004186159057576647147} a^{8} + \frac{2622824937343715063429993106530308}{14204799878995330706565548555124613} a^{7} + \frac{1802633721042450975082214304987019}{56819199515981322826262194220498452} a^{6} - \frac{16476830519706375173402545697674117}{56819199515981322826262194220498452} a^{5} + \frac{16170897606854709606559867544922771}{56819199515981322826262194220498452} a^{4} - \frac{61693330155047029700536416223443}{719230373620016744636230306588588} a^{3} + \frac{6897894885445862321103824731601798}{14204799878995330706565548555124613} a^{2} + \frac{13577007286142508848593195107192841}{56819199515981322826262194220498452} a + \frac{1686223874505433921694866816098}{5415478413646713955991440547131}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{364364098457571}{207400872804292820803} a^{17} + \frac{8470847911051943}{829603491217171283212} a^{16} + \frac{12655034244460129}{829603491217171283212} a^{15} - \frac{9114402344017391}{48800205365715957836} a^{14} + \frac{13541076340459015}{414801745608585641606} a^{13} + \frac{759692914170821617}{414801745608585641606} a^{12} - \frac{2238504867100437601}{829603491217171283212} a^{11} - \frac{5969365566453214405}{829603491217171283212} a^{10} + \frac{11203687387169003263}{829603491217171283212} a^{9} + \frac{12068981103106149881}{414801745608585641606} a^{8} - \frac{71593234695704845825}{829603491217171283212} a^{7} + \frac{2985105994737669467}{829603491217171283212} a^{6} + \frac{49542960340331085291}{829603491217171283212} a^{5} + \frac{9342765837580502723}{48800205365715957836} a^{4} - \frac{254093235546166278319}{414801745608585641606} a^{3} + \frac{156296732622930161005}{414801745608585641606} a^{2} - \frac{590252664696434040735}{829603491217171283212} a + \frac{3886921569721071737}{3400014308267095423} \) (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:  \( 24722362.03581395 \) (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

\(\Q(\sqrt{-3}) \), 3.3.316.1, 6.0.2696112.3, 6.0.2696112.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$[\ ]_{3}^{6}$
2.12.16.3$x^{12} - 30 x^{10} - 5 x^{8} + 19 x^{4} + 30 x^{2} + 1$$6$$2$$16$$C_6\times S_3$$[2]_{3}^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$79$79.3.2.2$x^{3} + 158$$3$$1$$2$$C_3$$[\ ]_{3}$
79.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
79.6.3.1$x^{6} - 158 x^{4} + 6241 x^{2} - 7888624$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
79.6.5.3$x^{6} - 1264$$6$$1$$5$$C_6$$[\ ]_{6}$