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Properties

Label 18.0.71155975251...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{33}\cdot 5^{9}$
Root discriminant $31.03$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1536, -6912, 16128, -27648, 33408, -23328, 7536, 3744, -8190, 6561, -3231, 828, 288, -432, 270, -126, 42, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 42*x^16 - 126*x^15 + 270*x^14 - 432*x^13 + 288*x^12 + 828*x^11 - 3231*x^10 + 6561*x^9 - 8190*x^8 + 3744*x^7 + 7536*x^6 - 23328*x^5 + 33408*x^4 - 27648*x^3 + 16128*x^2 - 6912*x + 1536)
 
gp: K = bnfinit(x^18 - 9*x^17 + 42*x^16 - 126*x^15 + 270*x^14 - 432*x^13 + 288*x^12 + 828*x^11 - 3231*x^10 + 6561*x^9 - 8190*x^8 + 3744*x^7 + 7536*x^6 - 23328*x^5 + 33408*x^4 - 27648*x^3 + 16128*x^2 - 6912*x + 1536, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 42 x^{16} - 126 x^{15} + 270 x^{14} - 432 x^{13} + 288 x^{12} + 828 x^{11} - 3231 x^{10} + 6561 x^{9} - 8190 x^{8} + 3744 x^{7} + 7536 x^{6} - 23328 x^{5} + 33408 x^{4} - 27648 x^{3} + 16128 x^{2} - 6912 x + 1536 \)

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:  \(-711559752519106944000000000=-\,2^{16}\cdot 3^{33}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.03$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{16} a^{5} - \frac{7}{16} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{3}{8} a^{7} + \frac{1}{32} a^{6} - \frac{7}{32} a^{5} + \frac{3}{16} a^{4} - \frac{3}{8} a^{3}$, $\frac{1}{64} a^{15} - \frac{1}{64} a^{14} - \frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{3}{16} a^{8} - \frac{31}{64} a^{7} - \frac{7}{64} a^{6} + \frac{3}{32} a^{5} - \frac{3}{16} a^{4}$, $\frac{1}{9088} a^{16} - \frac{61}{9088} a^{15} + \frac{43}{4544} a^{14} - \frac{47}{4544} a^{13} - \frac{149}{4544} a^{12} + \frac{13}{1136} a^{11} + \frac{3}{568} a^{10} - \frac{1089}{2272} a^{9} + \frac{433}{9088} a^{8} - \frac{4211}{9088} a^{7} + \frac{723}{4544} a^{6} + \frac{13}{568} a^{5} - \frac{17}{284} a^{4} - \frac{13}{284} a^{3} + \frac{21}{71} a^{2} + \frac{37}{142} a + \frac{3}{71}$, $\frac{1}{97980788672225814960896} a^{17} - \frac{1651549284990315311}{97980788672225814960896} a^{16} - \frac{15840256627972136111}{12247598584028226870112} a^{15} - \frac{280950424210513332613}{48990394336112907480448} a^{14} + \frac{222554911461406387485}{48990394336112907480448} a^{13} + \frac{93907937162867931895}{24495197168056453740224} a^{12} + \frac{259510645053222040285}{3061899646007056717528} a^{11} - \frac{756250518749242912397}{24495197168056453740224} a^{10} - \frac{371870751312407359489}{1380011108059518520576} a^{9} + \frac{28816760246344222641819}{97980788672225814960896} a^{8} + \frac{11593212493477328963601}{24495197168056453740224} a^{7} + \frac{11275371055231477048437}{24495197168056453740224} a^{6} - \frac{4721551557583026767789}{12247598584028226870112} a^{5} - \frac{2111088398149103097235}{6123799292014113435056} a^{4} - \frac{605408007280139634571}{1530949823003528358764} a^{3} - \frac{3443232709837063354}{382737455750882089691} a^{2} - \frac{117989166018921660419}{382737455750882089691} a - \frac{92480378567011827707}{382737455750882089691}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 2701291.1448069788 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

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 $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.108.1, 6.0.4374000.1, 9.1.11019960576.1

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

Sibling fields

Degree 18 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 R $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
3Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$