Properties

Label 18.0.92836019722...0243.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 31^{6}$
Root discriminant $11.32$
Ramified primes $3, 31$
Class number $1$
Class group Trivial
Galois Group $C_3.S_3^2$ (as 18T57)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 9, 39, 103, 201, 303, 342, 231, -6, -205, -219, -75, 54, 69, 21, -11, -9, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^16 - 11*x^15 + 21*x^14 + 69*x^13 + 54*x^12 - 75*x^11 - 219*x^10 - 205*x^9 - 6*x^8 + 231*x^7 + 342*x^6 + 303*x^5 + 201*x^4 + 103*x^3 + 39*x^2 + 9*x + 1)
gp: K = bnfinit(x^18 - 9*x^16 - 11*x^15 + 21*x^14 + 69*x^13 + 54*x^12 - 75*x^11 - 219*x^10 - 205*x^9 - 6*x^8 + 231*x^7 + 342*x^6 + 303*x^5 + 201*x^4 + 103*x^3 + 39*x^2 + 9*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 9 x^{16} \) \(\mathstrut -\mathstrut 11 x^{15} \) \(\mathstrut +\mathstrut 21 x^{14} \) \(\mathstrut +\mathstrut 69 x^{13} \) \(\mathstrut +\mathstrut 54 x^{12} \) \(\mathstrut -\mathstrut 75 x^{11} \) \(\mathstrut -\mathstrut 219 x^{10} \) \(\mathstrut -\mathstrut 205 x^{9} \) \(\mathstrut -\mathstrut 6 x^{8} \) \(\mathstrut +\mathstrut 231 x^{7} \) \(\mathstrut +\mathstrut 342 x^{6} \) \(\mathstrut +\mathstrut 303 x^{5} \) \(\mathstrut +\mathstrut 201 x^{4} \) \(\mathstrut +\mathstrut 103 x^{3} \) \(\mathstrut +\mathstrut 39 x^{2} \) \(\mathstrut +\mathstrut 9 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-9283601972222640243=-\,3^{21}\cdot 31^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.32$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 31$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{179} a^{15} - \frac{1}{179} a^{14} - \frac{39}{179} a^{13} + \frac{29}{179} a^{12} - \frac{22}{179} a^{11} + \frac{4}{179} a^{10} + \frac{41}{179} a^{9} + \frac{62}{179} a^{8} - \frac{61}{179} a^{7} - \frac{74}{179} a^{6} - \frac{80}{179} a^{5} - \frac{40}{179} a^{4} + \frac{70}{179} a^{3} - \frac{65}{179} a^{2} + \frac{12}{179} a + \frac{6}{179}$, $\frac{1}{179} a^{16} - \frac{40}{179} a^{14} - \frac{10}{179} a^{13} + \frac{7}{179} a^{12} - \frac{18}{179} a^{11} + \frac{45}{179} a^{10} - \frac{76}{179} a^{9} + \frac{1}{179} a^{8} + \frac{44}{179} a^{7} + \frac{25}{179} a^{6} + \frac{59}{179} a^{5} + \frac{30}{179} a^{4} + \frac{5}{179} a^{3} - \frac{53}{179} a^{2} + \frac{18}{179} a + \frac{6}{179}$, $\frac{1}{119393} a^{17} + \frac{122}{119393} a^{16} + \frac{201}{119393} a^{15} - \frac{835}{119393} a^{14} + \frac{32885}{119393} a^{13} + \frac{13374}{119393} a^{12} - \frac{41821}{119393} a^{11} + \frac{17655}{119393} a^{10} - \frac{40739}{119393} a^{9} + \frac{788}{119393} a^{8} - \frac{320}{667} a^{7} + \frac{68}{5191} a^{6} + \frac{28402}{119393} a^{5} - \frac{29066}{119393} a^{4} + \frac{12594}{119393} a^{3} + \frac{47160}{119393} a^{2} - \frac{59346}{119393} a - \frac{47942}{119393}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( -\frac{100290}{119393} a^{17} + \frac{26081}{119393} a^{16} + \frac{889562}{119393} a^{15} + \frac{856728}{119393} a^{14} - \frac{2230237}{119393} a^{13} - \frac{6188582}{119393} a^{12} - \frac{4102693}{119393} a^{11} + \frac{7808820}{119393} a^{10} + \frac{19646959}{119393} a^{9} + \frac{16689982}{119393} a^{8} - \frac{1534437}{119393} a^{7} - \frac{948902}{5191} a^{6} - \frac{29930698}{119393} a^{5} - \frac{25103525}{119393} a^{4} - \frac{15893659}{119393} a^{3} - \frac{7431791}{119393} a^{2} - \frac{2550686}{119393} a - \frac{338506}{119393} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{123655}{119393} a^{17} + \frac{9709}{119393} a^{16} - \frac{1129664}{119393} a^{15} - \frac{1459054}{119393} a^{14} + \frac{2683835}{119393} a^{13} + \frac{8957979}{119393} a^{12} + \frac{6810711}{119393} a^{11} - \frac{10104377}{119393} a^{10} - \frac{28484415}{119393} a^{9} - \frac{25505969}{119393} a^{8} + \frac{1440696}{119393} a^{7} + \frac{1355450}{5191} a^{6} + \frac{43061683}{119393} a^{5} + \frac{36019284}{119393} a^{4} + \frac{22400665}{119393} a^{3} + \frac{11030318}{119393} a^{2} + \frac{3813657}{119393} a + \frac{613616}{119393} \),  \( \frac{14496}{119393} a^{17} - \frac{117764}{119393} a^{16} - \frac{59123}{119393} a^{15} + \frac{892335}{119393} a^{14} + \frac{927859}{119393} a^{13} - \frac{2135116}{119393} a^{12} - \frac{5551023}{119393} a^{11} - \frac{2743724}{119393} a^{10} + \frac{8303401}{119393} a^{9} + \frac{16131201}{119393} a^{8} + \frac{9206679}{119393} a^{7} - \frac{283022}{5191} a^{6} - \frac{18240145}{119393} a^{5} - \frac{18746206}{119393} a^{4} - \frac{12520202}{119393} a^{3} - \frac{7199883}{119393} a^{2} - \frac{2700374}{119393} a - \frac{480829}{119393} \),  \( \frac{113658}{119393} a^{17} - \frac{19330}{119393} a^{16} - \frac{1037344}{119393} a^{15} - \frac{1049526}{119393} a^{14} + \frac{2710797}{119393} a^{13} + \frac{7342777}{119393} a^{12} + \frac{4420449}{119393} a^{11} - \frac{9761703}{119393} a^{10} - \frac{22990752}{119393} a^{9} - \frac{18055945}{119393} a^{8} + \frac{3843886}{119393} a^{7} + \frac{1125473}{5191} a^{6} + \frac{33382615}{119393} a^{5} + \frac{26680205}{119393} a^{4} + \frac{16260965}{119393} a^{3} + \frac{7623707}{119393} a^{2} + \frac{2487130}{119393} a + \frac{461529}{119393} \),  \( \frac{86126}{119393} a^{17} - \frac{34563}{119393} a^{16} - \frac{797724}{119393} a^{15} - \frac{575558}{119393} a^{14} + \frac{2319917}{119393} a^{13} + \frac{4998652}{119393} a^{12} + \frac{1668092}{119393} a^{11} - \frac{8395563}{119393} a^{10} - \frac{14915101}{119393} a^{9} - \frac{8330125}{119393} a^{8} + \frac{6135202}{119393} a^{7} + \frac{751843}{5191} a^{6} + \frac{18963665}{119393} a^{5} + \frac{14081765}{119393} a^{4} + \frac{8605747}{119393} a^{3} + \frac{3609461}{119393} a^{2} + \frac{1002578}{119393} a + \frac{54830}{119393} \),  \( \frac{2396}{5191} a^{17} - \frac{3372}{5191} a^{16} - \frac{19060}{5191} a^{15} + \frac{2457}{5191} a^{14} + \frac{65957}{5191} a^{13} + \frac{79550}{5191} a^{12} - \frac{42104}{5191} a^{11} - \frac{223437}{5191} a^{10} - \frac{218639}{5191} a^{9} + \frac{20999}{5191} a^{8} + \frac{256241}{5191} a^{7} + \frac{316354}{5191} a^{6} + \frac{199950}{5191} a^{5} + \frac{73255}{5191} a^{4} + \frac{9859}{5191} a^{3} - \frac{23208}{5191} a^{2} - \frac{14948}{5191} a - \frac{4904}{5191} \),  \( \frac{110179}{119393} a^{17} - \frac{38232}{119393} a^{16} - \frac{923550}{119393} a^{15} - \frac{971307}{119393} a^{14} + \frac{2257497}{119393} a^{13} + \frac{6812952}{119393} a^{12} + \frac{4857882}{119393} a^{11} - \frac{8013879}{119393} a^{10} - \frac{21538542}{119393} a^{9} - \frac{19529430}{119393} a^{8} + \frac{396361}{119393} a^{7} + \frac{1023722}{5191} a^{6} + \frac{33739616}{119393} a^{5} + \frac{29360957}{119393} a^{4} + \frac{18614178}{119393} a^{3} + \frac{9071451}{119393} a^{2} + \frac{3221050}{119393} a + \frac{607800}{119393} \),  \( \frac{212347}{119393} a^{17} + \frac{28735}{119393} a^{16} - \frac{2009921}{119393} a^{15} - \frac{2513057}{119393} a^{14} + \frac{4995821}{119393} a^{13} + \frac{15615994}{119393} a^{12} + \frac{10812246}{119393} a^{11} - \frac{19144172}{119393} a^{10} - \frac{49233130}{119393} a^{9} - \frac{40394374}{119393} a^{8} + \frac{6859180}{119393} a^{7} + \frac{2367418}{5191} a^{6} + \frac{70752153}{119393} a^{5} + \frac{56854341}{119393} a^{4} + \frac{35068496}{119393} a^{3} + \frac{16844965}{119393} a^{2} + \frac{5088638}{119393} a + \frac{730784}{119393} \),  \( \frac{133286}{119393} a^{17} - \frac{29916}{119393} a^{16} - \frac{1222180}{119393} a^{15} - \frac{1153434}{119393} a^{14} + \frac{3281957}{119393} a^{13} + \frac{8473694}{119393} a^{12} + \frac{4580834}{119393} a^{11} - \frac{12084370}{119393} a^{10} - \frac{26412608}{119393} a^{9} - \frac{19048640}{119393} a^{8} + \frac{6709343}{119393} a^{7} + \frac{1324334}{5191} a^{6} + \frac{36611420}{119393} a^{5} + \frac{27865794}{119393} a^{4} + \frac{16410536}{119393} a^{3} + \frac{7529543}{119393} a^{2} + \frac{2355907}{119393} a + \frac{341494}{119393} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 296.277259881 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_3.S_3^2$ (as 18T57):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 108
The 11 conjugacy class representatives for $C_3.S_3^2$
Character table for $C_3.S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.31.1, 6.0.25947.1, 9.1.586376253.1 x3

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

Sibling fields

Degree 9 siblings: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.12.14.5$x^{12} - 12 x^{11} - 3 x^{10} - 9 x^{7} + 9 x^{5} + 9 x^{2} + 9$$6$$2$$14$$S_3^2$$[3/2, 3/2]_{2}^{2}$
$31$31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$