Properties

Label 18.0.54609123358...1107.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 23^{6}\cdot 37^{4}$
Root discriminant $10.99$
Ramified primes $3, 23, 37$
Class number $1$
Class group Trivial
Galois Group $C_3\wr S_3:C_2$ (as 18T137)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 16, -10, 0, -32, 63, -28, 3, -35, 62, -56, 57, -84, 100, -76, 35, -9, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 35*x^16 - 76*x^15 + 100*x^14 - 84*x^13 + 57*x^12 - 56*x^11 + 62*x^10 - 35*x^9 + 3*x^8 - 28*x^7 + 63*x^6 - 32*x^5 - 10*x^3 + 16*x^2 - 7*x + 1)
gp: K = bnfinit(x^18 - 9*x^17 + 35*x^16 - 76*x^15 + 100*x^14 - 84*x^13 + 57*x^12 - 56*x^11 + 62*x^10 - 35*x^9 + 3*x^8 - 28*x^7 + 63*x^6 - 32*x^5 - 10*x^3 + 16*x^2 - 7*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 9 x^{17} \) \(\mathstrut +\mathstrut 35 x^{16} \) \(\mathstrut -\mathstrut 76 x^{15} \) \(\mathstrut +\mathstrut 100 x^{14} \) \(\mathstrut -\mathstrut 84 x^{13} \) \(\mathstrut +\mathstrut 57 x^{12} \) \(\mathstrut -\mathstrut 56 x^{11} \) \(\mathstrut +\mathstrut 62 x^{10} \) \(\mathstrut -\mathstrut 35 x^{9} \) \(\mathstrut +\mathstrut 3 x^{8} \) \(\mathstrut -\mathstrut 28 x^{7} \) \(\mathstrut +\mathstrut 63 x^{6} \) \(\mathstrut -\mathstrut 32 x^{5} \) \(\mathstrut -\mathstrut 10 x^{3} \) \(\mathstrut +\mathstrut 16 x^{2} \) \(\mathstrut -\mathstrut 7 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-5460912335827351107=-\,3^{9}\cdot 23^{6}\cdot 37^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.99$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 23, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{45} a^{16} + \frac{7}{45} a^{15} - \frac{1}{45} a^{14} - \frac{1}{15} a^{13} + \frac{4}{9} a^{12} + \frac{1}{9} a^{11} + \frac{4}{15} a^{10} - \frac{19}{45} a^{9} + \frac{7}{45} a^{8} + \frac{1}{5} a^{7} + \frac{11}{45} a^{6} - \frac{14}{45} a^{5} + \frac{11}{45} a^{4} + \frac{11}{45} a^{3} - \frac{4}{15} a^{2} + \frac{1}{3} a + \frac{7}{45}$, $\frac{1}{45} a^{17} - \frac{1}{9} a^{15} + \frac{4}{45} a^{14} - \frac{4}{45} a^{13} + \frac{22}{45} a^{11} - \frac{13}{45} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{7}{45} a^{7} - \frac{1}{45} a^{6} + \frac{19}{45} a^{5} - \frac{7}{15} a^{4} + \frac{1}{45} a^{3} + \frac{1}{5} a^{2} - \frac{8}{45} a - \frac{4}{45}$

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:  \( \frac{13946}{45} a^{17} - \frac{118541}{45} a^{16} + \frac{47652}{5} a^{15} - \frac{169138}{9} a^{14} + \frac{972514}{45} a^{13} - \frac{137306}{9} a^{12} + \frac{452902}{45} a^{11} - \frac{111034}{9} a^{10} + \frac{195848}{15} a^{9} - \frac{195172}{45} a^{8} - \frac{54986}{45} a^{7} - \frac{139324}{15} a^{6} + \frac{223166}{15} a^{5} - \frac{112507}{45} a^{4} - \frac{11062}{9} a^{3} - \frac{18536}{5} a^{2} + \frac{139697}{45} a - \frac{28111}{45} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{3667}{45} a^{17} - \frac{31532}{45} a^{16} + \frac{115706}{45} a^{15} - \frac{15490}{3} a^{14} + \frac{273713}{45} a^{13} - \frac{39713}{9} a^{12} + \frac{43213}{15} a^{11} - \frac{30701}{9} a^{10} + \frac{165973}{45} a^{9} - \frac{20678}{15} a^{8} - \frac{13732}{45} a^{7} - \frac{108134}{45} a^{6} + \frac{187706}{45} a^{5} - \frac{42409}{45} a^{4} - \frac{1124}{3} a^{3} - \frac{4812}{5} a^{2} + \frac{41434}{45} a - \frac{3049}{15} \),  \( \frac{10459}{45} a^{17} - \frac{88792}{45} a^{16} + \frac{106912}{15} a^{15} - \frac{631132}{45} a^{14} + \frac{144730}{9} a^{13} - \frac{101812}{9} a^{12} + \frac{336293}{45} a^{11} - \frac{414121}{45} a^{10} + \frac{48567}{5} a^{9} - \frac{142874}{45} a^{8} - \frac{41611}{45} a^{7} - \frac{34904}{5} a^{6} + \frac{55396}{5} a^{5} - \frac{80006}{45} a^{4} - \frac{40933}{45} a^{3} - \frac{8372}{3} a^{2} + \frac{103213}{45} a - \frac{4096}{9} \),  \( \frac{8129}{45} a^{17} - \frac{68339}{45} a^{16} + \frac{81304}{15} a^{15} - \frac{94519}{9} a^{14} + \frac{530806}{45} a^{13} - \frac{72971}{9} a^{12} + \frac{243883}{45} a^{11} - \frac{61768}{9} a^{10} + \frac{106517}{15} a^{9} - \frac{92998}{45} a^{8} - \frac{32579}{45} a^{7} - \frac{27432}{5} a^{6} + \frac{121999}{15} a^{5} - \frac{40978}{45} a^{4} - \frac{5281}{9} a^{3} - \frac{32497}{15} a^{2} + \frac{72263}{45} a - \frac{13249}{45} \),  \( \frac{2944}{15} a^{17} - \frac{8343}{5} a^{16} + \frac{90587}{15} a^{15} - 11917 a^{14} + \frac{205826}{15} a^{13} - \frac{29122}{3} a^{12} + \frac{96103}{15} a^{11} - 7831 a^{10} + \frac{41457}{5} a^{9} - \frac{41648}{15} a^{8} - \frac{11399}{15} a^{7} - \frac{88138}{15} a^{6} + \frac{141502}{15} a^{5} - \frac{24308}{15} a^{4} - \frac{2305}{3} a^{3} - \frac{35036}{15} a^{2} + \frac{29648}{15} a - \frac{6064}{15} \),  \( \frac{2944}{15} a^{17} - \frac{25019}{15} a^{16} + \frac{30169}{5} a^{15} - \frac{35698}{3} a^{14} + \frac{68457}{5} a^{13} - 9680 a^{12} + \frac{95918}{15} a^{11} - 7822 a^{10} + \frac{124081}{15} a^{9} - \frac{41363}{15} a^{8} - \frac{11404}{15} a^{7} - \frac{88183}{15} a^{6} + \frac{141152}{15} a^{5} - \frac{7991}{5} a^{4} - \frac{2282}{3} a^{3} - \frac{35066}{15} a^{2} + \frac{29533}{15} a - \frac{6004}{15} \),  \( \frac{244}{5} a^{17} - \frac{18893}{45} a^{16} + \frac{69364}{45} a^{15} - \frac{139363}{45} a^{14} + 3650 a^{13} - \frac{23837}{9} a^{12} + \frac{77792}{45} a^{11} - \frac{10236}{5} a^{10} + \frac{99662}{45} a^{9} - \frac{37301}{45} a^{8} - \frac{2758}{15} a^{7} - \frac{64819}{45} a^{6} + \frac{112771}{45} a^{5} - \frac{25549}{45} a^{4} - \frac{10222}{45} a^{3} - \frac{1738}{3} a^{2} + \frac{2768}{5} a - \frac{1096}{9} \),  \( \frac{1201}{45} a^{17} - \frac{9608}{45} a^{16} + \frac{32129}{45} a^{15} - \frac{6307}{5} a^{14} + \frac{11074}{9} a^{13} - \frac{6293}{9} a^{12} + \frac{7784}{15} a^{11} - \frac{36229}{45} a^{10} + \frac{32512}{45} a^{9} - \frac{59}{5} a^{8} - \frac{4984}{45} a^{7} - \frac{38549}{45} a^{6} + \frac{38396}{45} a^{5} + \frac{9581}{45} a^{4} + \frac{7}{5} a^{3} - \frac{965}{3} a^{2} + \frac{4777}{45} a \),  \( 8 a^{17} - \frac{3796}{45} a^{16} + \frac{17003}{45} a^{15} - \frac{41849}{45} a^{14} + \frac{20296}{15} a^{13} - \frac{10636}{9} a^{12} + \frac{6428}{9} a^{11} - \frac{9704}{15} a^{10} + \frac{37939}{45} a^{9} - \frac{26002}{45} a^{8} - \frac{143}{15} a^{7} - \frac{7376}{45} a^{6} + \frac{40859}{45} a^{5} - \frac{27836}{45} a^{4} - \frac{6656}{45} a^{3} - \frac{402}{5} a^{2} + 284 a - \frac{4027}{45} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 162.382523466 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_3\wr S_3:C_2$ (as 18T137):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 324
The 17 conjugacy class representatives for $C_3\wr S_3:C_2$
Character table for $C_3\wr S_3:C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.23.1, 6.0.14283.1, 9.1.449728821.1 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$37$37.3.2.3$x^{3} - 148$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.3$x^{3} - 148$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$
37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$