Properties

Label 18.0.10490638424...4432.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{8}\cdot 3^{9}\cdot 113^{6}$
Root discriminant $11.39$
Ramified primes $2, 3, 113$
Class number $1$
Class group Trivial
Galois Group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -2, -5, 17, -5, 28, -48, 49, -59, 64, -69, 56, -41, 31, -20, 11, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 11*x^16 - 20*x^15 + 31*x^14 - 41*x^13 + 56*x^12 - 69*x^11 + 64*x^10 - 59*x^9 + 49*x^8 - 48*x^7 + 28*x^6 - 5*x^5 + 17*x^4 - 5*x^3 - 2*x^2 - x + 1)
gp: K = bnfinit(x^18 - 4*x^17 + 11*x^16 - 20*x^15 + 31*x^14 - 41*x^13 + 56*x^12 - 69*x^11 + 64*x^10 - 59*x^9 + 49*x^8 - 48*x^7 + 28*x^6 - 5*x^5 + 17*x^4 - 5*x^3 - 2*x^2 - x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 4 x^{17} \) \(\mathstrut +\mathstrut 11 x^{16} \) \(\mathstrut -\mathstrut 20 x^{15} \) \(\mathstrut +\mathstrut 31 x^{14} \) \(\mathstrut -\mathstrut 41 x^{13} \) \(\mathstrut +\mathstrut 56 x^{12} \) \(\mathstrut -\mathstrut 69 x^{11} \) \(\mathstrut +\mathstrut 64 x^{10} \) \(\mathstrut -\mathstrut 59 x^{9} \) \(\mathstrut +\mathstrut 49 x^{8} \) \(\mathstrut -\mathstrut 48 x^{7} \) \(\mathstrut +\mathstrut 28 x^{6} \) \(\mathstrut -\mathstrut 5 x^{5} \) \(\mathstrut +\mathstrut 17 x^{4} \) \(\mathstrut -\mathstrut 5 x^{3} \) \(\mathstrut -\mathstrut 2 x^{2} \) \(\mathstrut -\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-10490638424730354432=-\,2^{8}\cdot 3^{9}\cdot 113^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.39$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{42} a^{15} + \frac{1}{14} a^{14} - \frac{1}{14} a^{13} + \frac{1}{14} a^{12} - \frac{2}{7} a^{11} + \frac{3}{14} a^{10} - \frac{4}{21} a^{9} - \frac{1}{21} a^{8} + \frac{19}{42} a^{7} + \frac{8}{21} a^{6} - \frac{3}{7} a^{5} + \frac{11}{42} a^{4} - \frac{17}{42} a^{3} + \frac{1}{42} a^{2} + \frac{3}{7} a - \frac{11}{42}$, $\frac{1}{42} a^{16} + \frac{1}{21} a^{14} - \frac{1}{21} a^{13} - \frac{1}{6} a^{12} + \frac{1}{14} a^{11} - \frac{1}{2} a^{10} - \frac{10}{21} a^{9} - \frac{1}{14} a^{8} - \frac{13}{42} a^{7} + \frac{3}{7} a^{6} - \frac{19}{42} a^{5} + \frac{1}{7} a^{4} - \frac{2}{21} a^{3} + \frac{1}{42} a^{2} - \frac{3}{14} a + \frac{19}{42}$, $\frac{1}{32298} a^{17} - \frac{3}{10766} a^{16} + \frac{4}{2307} a^{15} + \frac{719}{5383} a^{14} + \frac{1023}{10766} a^{13} + \frac{1535}{16149} a^{12} + \frac{4657}{16149} a^{11} - \frac{5113}{32298} a^{10} + \frac{853}{10766} a^{9} + \frac{2032}{16149} a^{8} + \frac{933}{10766} a^{7} + \frac{3009}{10766} a^{6} - \frac{14347}{32298} a^{5} - \frac{1887}{5383} a^{4} + \frac{10487}{32298} a^{3} + \frac{4540}{16149} a^{2} - \frac{1415}{5383} a - \frac{4819}{10766}$

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:  \( \frac{2147}{5383} a^{17} - \frac{42883}{32298} a^{16} + \frac{108499}{32298} a^{15} - \frac{23563}{4614} a^{14} + \frac{76833}{10766} a^{13} - \frac{42873}{5383} a^{12} + \frac{356539}{32298} a^{11} - \frac{62425}{5383} a^{10} + \frac{95129}{16149} a^{9} - \frac{137185}{32298} a^{8} + \frac{12227}{16149} a^{7} - \frac{56551}{16149} a^{6} - \frac{129631}{32298} a^{5} + \frac{228467}{32298} a^{4} + \frac{15083}{4614} a^{3} + \frac{15221}{5383} a^{2} - \frac{12381}{10766} a + \frac{3749}{16149} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{13772}{16149} a^{17} - \frac{18246}{5383} a^{16} + \frac{144497}{16149} a^{15} - \frac{245074}{16149} a^{14} + \frac{113719}{5383} a^{13} - \frac{56846}{2307} a^{12} + \frac{72964}{2307} a^{11} - \frac{608723}{16149} a^{10} + \frac{442991}{16149} a^{9} - \frac{281404}{16149} a^{8} + \frac{143968}{16149} a^{7} - \frac{228934}{16149} a^{6} + \frac{79183}{16149} a^{5} + \frac{198872}{16149} a^{4} + \frac{43369}{16149} a^{3} - \frac{2365}{2307} a^{2} - \frac{34742}{16149} a + \frac{5122}{5383} \),  \( \frac{15149}{16149} a^{17} - \frac{8162}{2307} a^{16} + \frac{7074}{769} a^{15} - \frac{80715}{5383} a^{14} + \frac{330749}{16149} a^{13} - \frac{122578}{5383} a^{12} + \frac{473063}{16149} a^{11} - \frac{539150}{16149} a^{10} + \frac{346282}{16149} a^{9} - \frac{199775}{16149} a^{8} + \frac{22867}{5383} a^{7} - \frac{178117}{16149} a^{6} + \frac{1285}{2307} a^{5} + \frac{207643}{16149} a^{4} + \frac{13052}{2307} a^{3} - \frac{11183}{16149} a^{2} - \frac{12993}{5383} a - \frac{9428}{16149} \),  \( \frac{357}{1538} a^{17} - \frac{22871}{32298} a^{16} + \frac{3842}{2307} a^{15} - \frac{30263}{16149} a^{14} + \frac{52105}{32298} a^{13} + \frac{632}{2307} a^{12} - \frac{9068}{16149} a^{11} + \frac{11561}{4614} a^{10} - \frac{224737}{32298} a^{9} + \frac{153814}{16149} a^{8} - \frac{363337}{32298} a^{7} + \frac{178751}{32298} a^{6} - \frac{230129}{32298} a^{5} + \frac{95180}{16149} a^{4} - \frac{6659}{10766} a^{3} + \frac{7324}{5383} a^{2} + \frac{10097}{5383} a + \frac{15649}{32298} \),  \( \frac{5315}{32298} a^{17} - \frac{18919}{16149} a^{16} + \frac{60385}{16149} a^{15} - \frac{44278}{5383} a^{14} + \frac{59281}{4614} a^{13} - \frac{540199}{32298} a^{12} + \frac{625501}{32298} a^{11} - \frac{390604}{16149} a^{10} + \frac{809539}{32298} a^{9} - \frac{79357}{4614} a^{8} + \frac{51330}{5383} a^{7} - \frac{56195}{10766} a^{6} + \frac{41610}{5383} a^{5} + \frac{7369}{16149} a^{4} - \frac{184813}{32298} a^{3} - \frac{10231}{10766} a^{2} - \frac{4813}{32298} a + \frac{3894}{5383} \),  \( \frac{6751}{32298} a^{17} - \frac{25385}{32298} a^{16} + \frac{18615}{10766} a^{15} - \frac{65119}{32298} a^{14} + \frac{14437}{16149} a^{13} + \frac{46391}{32298} a^{12} - \frac{45760}{16149} a^{11} + \frac{16958}{5383} a^{10} - \frac{254315}{32298} a^{9} + \frac{59929}{5383} a^{8} - \frac{52390}{5383} a^{7} + \frac{94921}{32298} a^{6} + \frac{2727}{10766} a^{5} + \frac{42529}{10766} a^{4} + \frac{41402}{16149} a^{3} - \frac{39035}{10766} a^{2} + \frac{1475}{2307} a - \frac{3338}{5383} \),  \( \frac{1235}{4614} a^{17} - \frac{33203}{32298} a^{16} + \frac{92699}{32298} a^{15} - \frac{56281}{10766} a^{14} + \frac{45154}{5383} a^{13} - \frac{364853}{32298} a^{12} + \frac{85718}{5383} a^{11} - \frac{317095}{16149} a^{10} + \frac{88979}{4614} a^{9} - \frac{303382}{16149} a^{8} + \frac{252617}{16149} a^{7} - \frac{490327}{32298} a^{6} + \frac{292877}{32298} a^{5} - \frac{113413}{32298} a^{4} + \frac{3708}{769} a^{3} - \frac{3793}{10766} a^{2} - \frac{5959}{16149} a - \frac{2147}{5383} \),  \( \frac{7137}{10766} a^{17} - \frac{11738}{5383} a^{16} + \frac{26811}{5383} a^{15} - \frac{32400}{5383} a^{14} + \frac{8243}{1538} a^{13} - \frac{18983}{10766} a^{12} + \frac{27035}{10766} a^{11} - \frac{6983}{5383} a^{10} - \frac{120131}{10766} a^{9} + \frac{183417}{10766} a^{8} - \frac{93743}{5383} a^{7} + \frac{43481}{10766} a^{6} - \frac{49926}{5383} a^{5} + \frac{11965}{769} a^{4} + \frac{3585}{1538} a^{3} - \frac{439}{10766} a^{2} - \frac{19769}{10766} a + \frac{3559}{5383} \),  \( \frac{6899}{16149} a^{17} - \frac{20631}{10766} a^{16} + \frac{26563}{4614} a^{15} - \frac{125561}{10766} a^{14} + \frac{627043}{32298} a^{13} - \frac{432815}{16149} a^{12} + \frac{381957}{10766} a^{11} - \frac{699578}{16149} a^{10} + \frac{102254}{2307} a^{9} - \frac{1343851}{32298} a^{8} + \frac{530552}{16149} a^{7} - \frac{150019}{5383} a^{6} + \frac{202989}{10766} a^{5} - \frac{294671}{32298} a^{4} + \frac{92527}{10766} a^{3} - \frac{35194}{16149} a^{2} + \frac{42121}{32298} a - \frac{27227}{16149} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 327.87771807 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.339.1, 6.0.344763.1, 9.3.623331504.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$113$113.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
113.12.6.1$x^{12} + 34629528 x^{6} - 18424351793 x^{2} + 299801052375696$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$