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Properties

Label 18.0.24452875343...4375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{9}\cdot 7^{15}\cdot 13^{15}\cdot 61^{6}$
Root discriminant $377.68$
Ramified primes $5, 7, 13, 61$
Class number $173715360$ (GRH)
Class group $[2, 2, 2, 2, 10857210]$ (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![153482306734184, 64347661656908, 34868873043404, 7963413486647, 2872304692608, 413878987297, 124438453933, 8469771305, 3289661823, -102865150, 48781375, -6792838, 995929, -128645, 23031, -1254, 256, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 256*x^16 - 1254*x^15 + 23031*x^14 - 128645*x^13 + 995929*x^12 - 6792838*x^11 + 48781375*x^10 - 102865150*x^9 + 3289661823*x^8 + 8469771305*x^7 + 124438453933*x^6 + 413878987297*x^5 + 2872304692608*x^4 + 7963413486647*x^3 + 34868873043404*x^2 + 64347661656908*x + 153482306734184)
 
gp: K = bnfinit(x^18 - 5*x^17 + 256*x^16 - 1254*x^15 + 23031*x^14 - 128645*x^13 + 995929*x^12 - 6792838*x^11 + 48781375*x^10 - 102865150*x^9 + 3289661823*x^8 + 8469771305*x^7 + 124438453933*x^6 + 413878987297*x^5 + 2872304692608*x^4 + 7963413486647*x^3 + 34868873043404*x^2 + 64347661656908*x + 153482306734184, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 256 x^{16} - 1254 x^{15} + 23031 x^{14} - 128645 x^{13} + 995929 x^{12} - 6792838 x^{11} + 48781375 x^{10} - 102865150 x^{9} + 3289661823 x^{8} + 8469771305 x^{7} + 124438453933 x^{6} + 413878987297 x^{5} + 2872304692608 x^{4} + 7963413486647 x^{3} + 34868873043404 x^{2} + 64347661656908 x + 153482306734184 \)

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:  \(-24452875343497320288961601897227861400802734375=-\,5^{9}\cdot 7^{15}\cdot 13^{15}\cdot 61^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $377.68$
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:  $5, 7, 13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{2}{9} a^{8} + \frac{2}{9} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} - \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{25182} a^{16} + \frac{1189}{25182} a^{15} - \frac{1511}{12591} a^{14} + \frac{22}{4197} a^{13} + \frac{419}{8394} a^{12} + \frac{775}{8394} a^{11} + \frac{251}{8394} a^{10} + \frac{419}{12591} a^{9} - \frac{887}{2798} a^{8} + \frac{4693}{12591} a^{7} - \frac{2261}{25182} a^{6} + \frac{7271}{25182} a^{5} - \frac{6857}{25182} a^{4} + \frac{2479}{25182} a^{3} - \frac{11}{4197} a^{2} + \frac{11335}{25182} a - \frac{4523}{12591}$, $\frac{1}{180499367810026540843059002496823774872065100208274331414827546491962385307106399652289931633386862732} a^{17} - \frac{1052316684465196685433149582727778890481705842979076328525433577055400597456607285388837710715193}{60166455936675513614353000832274591624021700069424777138275848830654128435702133217429977211128954244} a^{16} - \frac{2533221442400078311736663822265311488783098193470332668021830649295042122853268620082172733543796505}{90249683905013270421529501248411887436032550104137165707413773245981192653553199826144965816693431366} a^{15} - \frac{3441462394007207007732664378296245362357942063254113319625840870725643683874377780285887785500420709}{90249683905013270421529501248411887436032550104137165707413773245981192653553199826144965816693431366} a^{14} - \frac{1385536930029545119322697727069331006279739842559258019732571824833756457924714781605948619979975941}{20055485312225171204784333610758197208007233356474925712758616276884709478567377739143325737042984748} a^{13} + \frac{1107694574497884634795739652960560875017220448234759165341288584314895618271919172476455757772710837}{20055485312225171204784333610758197208007233356474925712758616276884709478567377739143325737042984748} a^{12} - \frac{892718354386677658417229799573106272445635231141784992099405957780901070941082075222215036797039263}{60166455936675513614353000832274591624021700069424777138275848830654128435702133217429977211128954244} a^{11} + \frac{4864029581534943679078841492436693351912887627075442763827397389213697373373695088214664991016320349}{45124841952506635210764750624205943718016275052068582853706886622990596326776599913072482908346715683} a^{10} + \frac{25097274859071989665057447700245907876057606495749860805541479683452280624293987816927770815013137191}{180499367810026540843059002496823774872065100208274331414827546491962385307106399652289931633386862732} a^{9} - \frac{2002141632005439173630843879977188684098395121551437744432233446460914938722719816260243934161522808}{45124841952506635210764750624205943718016275052068582853706886622990596326776599913072482908346715683} a^{8} - \frac{37433740238862143112217252359185505659652836469575113244113440176956286665050626604957107537898393121}{180499367810026540843059002496823774872065100208274331414827546491962385307106399652289931633386862732} a^{7} + \frac{57187210037746450200654906492173176250011312029292294786961112581193218298907704621999759510573955807}{180499367810026540843059002496823774872065100208274331414827546491962385307106399652289931633386862732} a^{6} - \frac{69205339004080276324688015047313602743022569647416593055233506781780762689658205892384790540886263241}{180499367810026540843059002496823774872065100208274331414827546491962385307106399652289931633386862732} a^{5} + \frac{11431627336979505647311655601869465369575158190588297726762670708291935611375416694508706020702815577}{60166455936675513614353000832274591624021700069424777138275848830654128435702133217429977211128954244} a^{4} + \frac{37444090803012524252804693296346076328198072513799052237947655259051355662056845870228614488140856439}{90249683905013270421529501248411887436032550104137165707413773245981192653553199826144965816693431366} a^{3} - \frac{73336455433674484975449810379647998251014073764660080547507234567217090806605281764064733994381900869}{180499367810026540843059002496823774872065100208274331414827546491962385307106399652289931633386862732} a^{2} + \frac{15217430538203412779266729274433864511423053435020082213124535797885722233554573688888037045268849199}{90249683905013270421529501248411887436032550104137165707413773245981192653553199826144965816693431366} a - \frac{7011157118079536447823088546988127972469658387706082927971796618093747164284254851717660884248438475}{45124841952506635210764750624205943718016275052068582853706886622990596326776599913072482908346715683}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{10857210}$, which has order $173715360$ (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:  \( 39072844.256274134 \) (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

\(\Q(\sqrt{-455}) \), 3.3.10309.1, 3.3.8281.2, Deg 6, 6.0.780040181375.2, 9.9.128895530697518221.2

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.2$x^{12} + 35 x^{6} + 441$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13Data not computed
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$