Properties

Label 18.0.25299902311...6912.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}$
Root discriminant $10.53$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois Group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 0, -9, 0, 18, -9, -6, 0, 0, 27, -63, 72, -21, -63, 105, -81, 36, -9, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 81*x^15 + 105*x^14 - 63*x^13 - 21*x^12 + 72*x^11 - 63*x^10 + 27*x^9 - 6*x^6 - 9*x^5 + 18*x^4 - 9*x^2 + 3)
gp: K = bnfinit(x^18 - 9*x^17 + 36*x^16 - 81*x^15 + 105*x^14 - 63*x^13 - 21*x^12 + 72*x^11 - 63*x^10 + 27*x^9 - 6*x^6 - 9*x^5 + 18*x^4 - 9*x^2 + 3, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 9 x^{17} \) \(\mathstrut +\mathstrut 36 x^{16} \) \(\mathstrut -\mathstrut 81 x^{15} \) \(\mathstrut +\mathstrut 105 x^{14} \) \(\mathstrut -\mathstrut 63 x^{13} \) \(\mathstrut -\mathstrut 21 x^{12} \) \(\mathstrut +\mathstrut 72 x^{11} \) \(\mathstrut -\mathstrut 63 x^{10} \) \(\mathstrut +\mathstrut 27 x^{9} \) \(\mathstrut -\mathstrut 6 x^{6} \) \(\mathstrut -\mathstrut 9 x^{5} \) \(\mathstrut +\mathstrut 18 x^{4} \) \(\mathstrut -\mathstrut 9 x^{2} \) \(\mathstrut +\mathstrut 3 \)

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:  \(-2529990231179046912=-\,2^{12}\cdot 3^{31}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.53$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{159031} a^{17} + \frac{66377}{159031} a^{16} + \frac{72610}{159031} a^{15} + \frac{57769}{159031} a^{14} + \frac{20374}{159031} a^{13} - \frac{10354}{159031} a^{12} - \frac{28683}{159031} a^{11} - \frac{71403}{159031} a^{10} + \frac{77396}{159031} a^{9} + \frac{37335}{159031} a^{8} + \frac{23175}{159031} a^{7} + \frac{29656}{159031} a^{6} - \frac{60570}{159031} a^{5} - \frac{60225}{159031} a^{4} - \frac{57492}{159031} a^{3} - \frac{78943}{159031} a^{2} - \frac{2433}{159031} a + \frac{58358}{159031}$

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:  \( \frac{89340}{159031} a^{17} - \frac{621903}{159031} a^{16} + \frac{1693220}{159031} a^{15} - \frac{1699924}{159031} a^{14} - \frac{1645976}{159031} a^{13} + \frac{6577238}{159031} a^{12} - \frac{7865236}{159031} a^{11} + \frac{4996444}{159031} a^{10} - \frac{1381488}{159031} a^{9} - \frac{1438573}{159031} a^{8} + \frac{1779252}{159031} a^{7} - \frac{625544}{159031} a^{6} + \frac{1455316}{159031} a^{5} - \frac{1755018}{159031} a^{4} - \frac{270104}{159031} a^{3} + \frac{575292}{159031} a^{2} + \frac{190188}{159031} a - \frac{127615}{159031} \) (order $18$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{153946}{159031} a^{17} - \frac{1176480}{159031} a^{16} + \frac{3864876}{159031} a^{15} - \frac{6545379}{159031} a^{14} + \frac{4698321}{159031} a^{13} + \frac{2237263}{159031} a^{12} - \frac{7134767}{159031} a^{11} + \frac{5582567}{159031} a^{10} - \frac{1547245}{159031} a^{9} - \frac{760616}{159031} a^{8} + \frac{792251}{159031} a^{7} + \frac{914814}{159031} a^{6} - \frac{362659}{159031} a^{5} - \frac{1957953}{159031} a^{4} + \frac{842997}{159031} a^{3} + \frac{1462190}{159031} a^{2} - \frac{191644}{159031} a - \frac{793739}{159031} \),  \( \frac{205995}{159031} a^{17} - \frac{1745575}{159031} a^{16} + \frac{6474578}{159031} a^{15} - \frac{13205117}{159031} a^{14} + \frac{14903923}{159031} a^{13} - \frac{6945822}{159031} a^{12} - \frac{3256462}{159031} a^{11} + \frac{6636476}{159031} a^{10} - \frac{4598691}{159031} a^{9} + \frac{1356413}{159031} a^{8} + \frac{300598}{159031} a^{7} + \frac{1561196}{159031} a^{6} - \frac{1453262}{159031} a^{5} - \frac{2266999}{159031} a^{4} + \frac{1405309}{159031} a^{3} + \frac{1600961}{159031} a^{2} - \frac{556247}{159031} a - \frac{492235}{159031} \),  \( \frac{266986}{159031} a^{17} - \frac{2275228}{159031} a^{16} + \frac{8403203}{159031} a^{15} - \frac{16805556}{159031} a^{14} + \frac{17887912}{159031} a^{13} - \frac{6139380}{159031} a^{12} - \frac{7137059}{159031} a^{11} + \frac{9940658}{159031} a^{10} - \frac{6087707}{159031} a^{9} + \frac{1449540}{159031} a^{8} + \frac{935619}{159031} a^{7} + \frac{1650729}{159031} a^{6} - \frac{1388002}{159031} a^{5} - \frac{3583215}{159031} a^{4} + \frac{1862349}{159031} a^{3} + \frac{2406159}{159031} a^{2} - \frac{571427}{159031} a - \frac{770330}{159031} \),  \( \frac{556695}{159031} a^{17} - \frac{4038796}{159031} a^{16} + \frac{12323943}{159031} a^{15} - \frac{18460054}{159031} a^{14} + \frac{9077777}{159031} a^{13} + \frac{11985890}{159031} a^{12} - \frac{21326253}{159031} a^{11} + \frac{13782031}{159031} a^{10} - \frac{3165168}{159031} a^{9} - \frac{3847402}{159031} a^{8} + \frac{3038339}{159031} a^{7} + \frac{2883306}{159031} a^{6} + \frac{1917090}{159031} a^{5} - \frac{6402195}{159031} a^{4} - \frac{679221}{159031} a^{3} + \frac{3746992}{159031} a^{2} + \frac{823247}{159031} a - \frac{836180}{159031} \),  \( \frac{471110}{159031} a^{17} - \frac{3531866}{159031} a^{16} + \frac{11338263}{159031} a^{15} - \frac{18823222}{159031} a^{14} + \frac{13914832}{159031} a^{13} + \frac{3424574}{159031} a^{12} - \frac{15251036}{159031} a^{11} + \frac{13882580}{159031} a^{10} - \frac{6959360}{159031} a^{9} - \frac{254812}{159031} a^{8} + \frac{2086410}{159031} a^{7} + \frac{1796089}{159031} a^{6} + \frac{1389940}{159031} a^{5} - \frac{5127063}{159031} a^{4} + \frac{149614}{159031} a^{3} + \frac{2538395}{159031} a^{2} + \frac{402880}{159031} a - \frac{400931}{159031} \),  \( \frac{178216}{159031} a^{17} - \frac{1190720}{159031} a^{16} + \frac{3409972}{159031} a^{15} - \frac{5236797}{159031} a^{14} + \frac{4906953}{159031} a^{13} - \frac{4623670}{159031} a^{12} + \frac{6325145}{159031} a^{11} - \frac{5857668}{159031} a^{10} + \frac{2832371}{159031} a^{9} - \frac{480742}{159031} a^{8} - \frac{1946673}{159031} a^{7} + \frac{2959031}{159031} a^{6} - \frac{154964}{159031} a^{5} + \frac{579714}{159031} a^{4} - \frac{2012407}{159031} a^{3} + \frac{89789}{159031} a^{2} + \frac{873164}{159031} a + \frac{497083}{159031} \),  \( \frac{70412}{159031} a^{17} - \frac{978921}{159031} a^{16} + \frac{5016693}{159031} a^{15} - \frac{13422694}{159031} a^{14} + \frac{20152374}{159031} a^{13} - \frac{15314720}{159031} a^{12} + \frac{1656614}{159031} a^{11} + \frac{6657300}{159031} a^{10} - \frac{6905489}{159031} a^{9} + \frac{4184396}{159031} a^{8} - \frac{814146}{159031} a^{7} + \frac{1015428}{159031} a^{6} - \frac{3301133}{159031} a^{5} - \frac{1273333}{159031} a^{4} + \frac{3347052}{159031} a^{3} + \frac{871182}{159031} a^{2} - \frac{1626319}{159031} a - \frac{893668}{159031} \),  \( \frac{600349}{159031} a^{17} - \frac{4439151}{159031} a^{16} + \frac{13825301}{159031} a^{15} - \frac{21229253}{159031} a^{14} + \frac{10773331}{159031} a^{13} + \frac{15139096}{159031} a^{12} - \frac{29513453}{159031} a^{11} + \frac{22509774}{159031} a^{10} - \frac{8540833}{159031} a^{9} - \frac{2661783}{159031} a^{8} + \frac{4395846}{159031} a^{7} + \frac{1542711}{159031} a^{6} + \frac{2797902}{159031} a^{5} - \frac{8272225}{159031} a^{4} + \frac{664501}{159031} a^{3} + \frac{3930071}{159031} a^{2} + \frac{527711}{159031} a - \frac{1111699}{159031} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 404.056392969 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, \(\Q(\zeta_{9})^+\), 6.0.34992.1, 6.0.314928.1 x2, \(\Q(\zeta_{9})\), 9.3.918330048.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.314928.1
Degree 9 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed