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Properties

Label 18.0.85364764011...4031.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{15}\cdot 11^{9}$
Root discriminant $87.22$
Ramified primes $3, 7, 11$
Class number $221616$ (GRH)
Class group $[2, 6, 18, 1026]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![843884056, -56209332, 428074110, -101345687, 133520847, -38234673, 35788472, -5411529, 7734366, -328269, 1098684, -9351, 93702, -111, 4482, -1, 108, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 108*x^16 - x^15 + 4482*x^14 - 111*x^13 + 93702*x^12 - 9351*x^11 + 1098684*x^10 - 328269*x^9 + 7734366*x^8 - 5411529*x^7 + 35788472*x^6 - 38234673*x^5 + 133520847*x^4 - 101345687*x^3 + 428074110*x^2 - 56209332*x + 843884056)
 
gp: K = bnfinit(x^18 + 108*x^16 - x^15 + 4482*x^14 - 111*x^13 + 93702*x^12 - 9351*x^11 + 1098684*x^10 - 328269*x^9 + 7734366*x^8 - 5411529*x^7 + 35788472*x^6 - 38234673*x^5 + 133520847*x^4 - 101345687*x^3 + 428074110*x^2 - 56209332*x + 843884056, 1)
 

Normalized defining polynomial

\( x^{18} + 108 x^{16} - x^{15} + 4482 x^{14} - 111 x^{13} + 93702 x^{12} - 9351 x^{11} + 1098684 x^{10} - 328269 x^{9} + 7734366 x^{8} - 5411529 x^{7} + 35788472 x^{6} - 38234673 x^{5} + 133520847 x^{4} - 101345687 x^{3} + 428074110 x^{2} - 56209332 x + 843884056 \)

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:  \(-85364764011113444925996300478214031=-\,3^{27}\cdot 7^{15}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.22$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(693=3^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{693}(1,·)$, $\chi_{693}(131,·)$, $\chi_{693}(395,·)$, $\chi_{693}(461,·)$, $\chi_{693}(463,·)$, $\chi_{693}(593,·)$, $\chi_{693}(67,·)$, $\chi_{693}(100,·)$, $\chi_{693}(331,·)$, $\chi_{693}(164,·)$, $\chi_{693}(230,·)$, $\chi_{693}(529,·)$, $\chi_{693}(232,·)$, $\chi_{693}(298,·)$, $\chi_{693}(626,·)$, $\chi_{693}(562,·)$, $\chi_{693}(692,·)$, $\chi_{693}(362,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1508} a^{15} - \frac{16}{377} a^{13} + \frac{171}{754} a^{12} - \frac{86}{377} a^{11} + \frac{42}{377} a^{10} + \frac{61}{377} a^{9} + \frac{54}{377} a^{8} - \frac{18}{377} a^{7} - \frac{55}{754} a^{6} - \frac{101}{377} a^{5} + \frac{281}{754} a^{4} + \frac{353}{754} a^{3} + \frac{151}{754} a^{2} - \frac{31}{1508} a - \frac{359}{754}$, $\frac{1}{35705345780} a^{16} - \frac{10564973}{35705345780} a^{15} - \frac{4220599889}{35705345780} a^{14} + \frac{1239629567}{17852672890} a^{13} + \frac{7648740419}{35705345780} a^{12} - \frac{868777271}{17852672890} a^{11} + \frac{186375143}{2746565060} a^{10} - \frac{1882028004}{8926336445} a^{9} - \frac{5919805637}{35705345780} a^{8} - \frac{56985112}{307804705} a^{7} + \frac{12833557299}{35705345780} a^{6} + \frac{7108431723}{17852672890} a^{5} - \frac{43287089}{7141069156} a^{4} + \frac{443624053}{17852672890} a^{3} + \frac{1976905531}{8926336445} a^{2} + \frac{1935150495}{7141069156} a + \frac{4834867633}{17852672890}$, $\frac{1}{3986954482096186808501597244516577858044694699282082800} a^{17} + \frac{13192198312828282701878279535407159708754853}{1993477241048093404250798622258288929022347349641041400} a^{16} - \frac{123479464142175699392225863861780153394441607869139}{996738620524046702125399311129144464511173674820520700} a^{15} - \frac{331876153486622890354444699339461827575966594551820037}{3986954482096186808501597244516577858044694699282082800} a^{14} + \frac{577447674597836960193030121831693006646307042522073}{7667220157877282324041533162531880496239797498619390} a^{13} - \frac{77965480195651626905260522952031447609245345695327651}{3986954482096186808501597244516577858044694699282082800} a^{12} + \frac{124039461663456318714638923768357175498035894981739987}{498369310262023351062699655564572232255586837410260350} a^{11} + \frac{4682146888340547166626523882131350987390616970696149}{159478179283847472340063889780663114321787787971283312} a^{10} - \frac{348924949398848884998303790460489926243284874518956333}{1993477241048093404250798622258288929022347349641041400} a^{9} - \frac{198394529364748787702823710464854094993375020250716153}{797390896419237361700319448903315571608938939856416560} a^{8} + \frac{213267573374877542702103260517582871732840922764531819}{996738620524046702125399311129144464511173674820520700} a^{7} + \frac{1560545033779623737418877822593281685964609283096387627}{3986954482096186808501597244516577858044694699282082800} a^{6} + \frac{859115491881467282412242937884551851097933589072166167}{1993477241048093404250798622258288929022347349641041400} a^{5} - \frac{1139400280701751630421544255564197863259866231354771569}{3986954482096186808501597244516577858044694699282082800} a^{4} - \frac{910444779951436590183148724760426278679781381916038267}{3986954482096186808501597244516577858044694699282082800} a^{3} + \frac{776764772097365926708051164445986024964085046726014711}{3986954482096186808501597244516577858044694699282082800} a^{2} + \frac{26233101750093204990248366374979690058373428318673711}{249184655131011675531349827782286116127793418705130175} a + \frac{31510530830282432844685705618481633037777311312258287}{76672201578772823240415331625318804962397974986193900}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{18}\times C_{1026}$, which has order $221616$ (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:  \( 54408.4888887 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-231}) \), \(\Q(\zeta_{7})^+\), 3.3.3969.2, \(\Q(\zeta_{9})^+\), 3.3.3969.1, 6.0.603993159.1, 6.0.440311012911.4, 6.0.8985939039.8, 6.0.440311012911.5, 9.9.62523502209.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
7Data not computed
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$