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Properties

Label 18.0.24182281322...9063.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 83^{4}\cdot 181^{4}$
Root discriminant $42.90$
Ramified primes $7, 83, 181$
Class number $72$ (GRH)
Class group $[2, 2, 18]$ (GRH)
Galois group 18T201

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6889, -47725, 136210, -189258, 147567, -31451, 10025, 2590, 7661, -8361, 5655, -2492, 1478, -434, 217, -48, 22, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 22*x^16 - 48*x^15 + 217*x^14 - 434*x^13 + 1478*x^12 - 2492*x^11 + 5655*x^10 - 8361*x^9 + 7661*x^8 + 2590*x^7 + 10025*x^6 - 31451*x^5 + 147567*x^4 - 189258*x^3 + 136210*x^2 - 47725*x + 6889)
 
gp: K = bnfinit(x^18 - 3*x^17 + 22*x^16 - 48*x^15 + 217*x^14 - 434*x^13 + 1478*x^12 - 2492*x^11 + 5655*x^10 - 8361*x^9 + 7661*x^8 + 2590*x^7 + 10025*x^6 - 31451*x^5 + 147567*x^4 - 189258*x^3 + 136210*x^2 - 47725*x + 6889, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 22 x^{16} - 48 x^{15} + 217 x^{14} - 434 x^{13} + 1478 x^{12} - 2492 x^{11} + 5655 x^{10} - 8361 x^{9} + 7661 x^{8} + 2590 x^{7} + 10025 x^{6} - 31451 x^{5} + 147567 x^{4} - 189258 x^{3} + 136210 x^{2} - 47725 x + 6889 \)

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:  \(-241822813227897418142542819063=-\,7^{15}\cdot 83^{4}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.90$
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:  $7, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{53191541553785133001517504637326137800112441211} a^{17} - \frac{2764648350502012806956103529998515534068011219}{53191541553785133001517504637326137800112441211} a^{16} + \frac{24100089033223347674861077281510120984923676385}{53191541553785133001517504637326137800112441211} a^{15} - \frac{14267622140389216286057394040644845191295413185}{53191541553785133001517504637326137800112441211} a^{14} + \frac{8432490244792594772679280810564991106277933615}{53191541553785133001517504637326137800112441211} a^{13} + \frac{7688174087515229731139358750159568725828030750}{53191541553785133001517504637326137800112441211} a^{12} + \frac{18678047533829972094067774993099634276822853822}{53191541553785133001517504637326137800112441211} a^{11} + \frac{4235638308104854990265621042402947972447207214}{53191541553785133001517504637326137800112441211} a^{10} - \frac{19227807383803362960246582641324447099079198571}{53191541553785133001517504637326137800112441211} a^{9} - \frac{25552531547323405635863955166920370939836387712}{53191541553785133001517504637326137800112441211} a^{8} + \frac{12489066343492275724552268110531733722623016180}{53191541553785133001517504637326137800112441211} a^{7} + \frac{18566587740014837360957419020361116406279957398}{53191541553785133001517504637326137800112441211} a^{6} + \frac{4169287780059979695121778832783257017051674120}{53191541553785133001517504637326137800112441211} a^{5} + \frac{5415123422870613388691363827801723267682404703}{53191541553785133001517504637326137800112441211} a^{4} + \frac{84999230693314526757367234208426579857368070}{53191541553785133001517504637326137800112441211} a^{3} + \frac{24804802349531036649095030488101935103963926941}{53191541553785133001517504637326137800112441211} a^{2} - \frac{15154963475948926423148576969939332423096784}{79036465904584150076549041065863503417700507} a - \frac{149306435889979155808041657268923413561444768}{640861946431146180741174754666579973495330617}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{18}$, which has order $72$ (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:  \( -\frac{5295524372732271645035617123323294797831934}{53191541553785133001517504637326137800112441211} a^{17} + \frac{17231196584330831775880610443607661112547158}{53191541553785133001517504637326137800112441211} a^{16} - \frac{118008363853003733888226496283217423783527690}{53191541553785133001517504637326137800112441211} a^{15} + \frac{277645546195935852287524313463665181818970236}{53191541553785133001517504637326137800112441211} a^{14} - \frac{1162798789102610855619396696258439052348650111}{53191541553785133001517504637326137800112441211} a^{13} + \frac{2499890312737144319022611161598697310655942625}{53191541553785133001517504637326137800112441211} a^{12} - \frac{7934329025625663747904746973940576275131616385}{53191541553785133001517504637326137800112441211} a^{11} + \frac{14380515329356442653744810423345843616493408009}{53191541553785133001517504637326137800112441211} a^{10} - \frac{30198378227473488423484130427852891912735836033}{53191541553785133001517504637326137800112441211} a^{9} + \frac{47522032403749811053681690105415106801175689781}{53191541553785133001517504637326137800112441211} a^{8} - \frac{41054024546002297472488035331162351104374643312}{53191541553785133001517504637326137800112441211} a^{7} - \frac{17918628926870934371870791220547475418544372048}{53191541553785133001517504637326137800112441211} a^{6} - \frac{41279470910906503396001864543066173221885554369}{53191541553785133001517504637326137800112441211} a^{5} + \frac{192911673635713640216278984823499954911420368011}{53191541553785133001517504637326137800112441211} a^{4} - \frac{788790047986925616548810913367316237602647127882}{53191541553785133001517504637326137800112441211} a^{3} + \frac{1131774913737392499190036050776364682140088223980}{53191541553785133001517504637326137800112441211} a^{2} - \frac{995896942382476081872590431013493321961824664}{79036465904584150076549041065863503417700507} a + \frac{2196148461761171606980595672488680603595777132}{640861946431146180741174754666579973495330617} \) (order $14$)
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:  \( 1581190.17434 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T201:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 26 conjugacy class representatives for t18n201
Character table for t18n201 is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 9.9.26552265046321.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ $18$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ $18$

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
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$83$83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.6.4.1$x^{6} + 415 x^{3} + 55112$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$181$181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$