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Properties

Label 18.0.18572342613...0608.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 7^{12}\cdot 13^{15}$
Root discriminant $62.05$
Ramified primes $2, 7, 13$
Class number $2592$ (GRH)
Class group $[2, 2, 2, 18, 18]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![444536, -587872, 777688, -340356, 221722, -132222, 171273, -128674, 62720, -23298, 17101, -10402, 5146, -1946, 666, -162, 42, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 42*x^16 - 162*x^15 + 666*x^14 - 1946*x^13 + 5146*x^12 - 10402*x^11 + 17101*x^10 - 23298*x^9 + 62720*x^8 - 128674*x^7 + 171273*x^6 - 132222*x^5 + 221722*x^4 - 340356*x^3 + 777688*x^2 - 587872*x + 444536)
 
gp: K = bnfinit(x^18 - 6*x^17 + 42*x^16 - 162*x^15 + 666*x^14 - 1946*x^13 + 5146*x^12 - 10402*x^11 + 17101*x^10 - 23298*x^9 + 62720*x^8 - 128674*x^7 + 171273*x^6 - 132222*x^5 + 221722*x^4 - 340356*x^3 + 777688*x^2 - 587872*x + 444536, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 42 x^{16} - 162 x^{15} + 666 x^{14} - 1946 x^{13} + 5146 x^{12} - 10402 x^{11} + 17101 x^{10} - 23298 x^{9} + 62720 x^{8} - 128674 x^{7} + 171273 x^{6} - 132222 x^{5} + 221722 x^{4} - 340356 x^{3} + 777688 x^{2} - 587872 x + 444536 \)

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:  \(-185723426137096770687205679300608=-\,2^{18}\cdot 7^{12}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.05$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(364=2^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{364}(1,·)$, $\chi_{364}(261,·)$, $\chi_{364}(9,·)$, $\chi_{364}(207,·)$, $\chi_{364}(81,·)$, $\chi_{364}(23,·)$, $\chi_{364}(179,·)$, $\chi_{364}(155,·)$, $\chi_{364}(29,·)$, $\chi_{364}(95,·)$, $\chi_{364}(289,·)$, $\chi_{364}(165,·)$, $\chi_{364}(43,·)$, $\chi_{364}(303,·)$, $\chi_{364}(113,·)$, $\chi_{364}(51,·)$, $\chi_{364}(53,·)$, $\chi_{364}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{11} - \frac{1}{5} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{5} a^{5} + \frac{1}{10} a^{4} + \frac{3}{10} a^{3} + \frac{1}{10} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{11} + \frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{2} a^{7} + \frac{3}{10} a^{6} + \frac{3}{10} a^{5} - \frac{3}{10} a^{4} + \frac{3}{10} a^{3} + \frac{2}{5}$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{11} + \frac{1}{5} a^{10} + \frac{1}{10} a^{8} + \frac{3}{10} a^{7} + \frac{3}{10} a^{6} + \frac{1}{10} a^{5} + \frac{1}{10} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{10} a^{15} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{3}{10} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{10} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{924469886678097220} a^{16} + \frac{1616796332281923}{462234943339048610} a^{15} - \frac{11233555019496283}{462234943339048610} a^{14} + \frac{9454172176314762}{231117471669524305} a^{13} - \frac{2910758061108677}{462234943339048610} a^{12} - \frac{103589426233954037}{462234943339048610} a^{11} + \frac{15185138689429067}{231117471669524305} a^{10} + \frac{75259975632087943}{462234943339048610} a^{9} + \frac{199544557540536387}{924469886678097220} a^{8} + \frac{109357885467094279}{231117471669524305} a^{7} + \frac{159735850379331473}{462234943339048610} a^{6} + \frac{1844877761122959}{231117471669524305} a^{5} + \frac{39599376127364469}{924469886678097220} a^{4} + \frac{17043452303928972}{231117471669524305} a^{3} + \frac{174589521285633}{1276892108671405} a^{2} - \frac{55552561382551376}{231117471669524305} a + \frac{433954623074902}{1276892108671405}$, $\frac{1}{40682413003359742540366026580} a^{17} - \frac{6289788763}{40682413003359742540366026580} a^{16} + \frac{371718571731572990095908591}{20341206501679871270183013290} a^{15} + \frac{633447015172858676949620417}{20341206501679871270183013290} a^{14} + \frac{470463531194621222200520328}{10170603250839935635091506645} a^{13} - \frac{259013653596573564574346791}{10170603250839935635091506645} a^{12} - \frac{611470987343768971154977991}{20341206501679871270183013290} a^{11} - \frac{445951442540578806997395354}{10170603250839935635091506645} a^{10} - \frac{6952826813910008784546782807}{40682413003359742540366026580} a^{9} + \frac{4227210045206766400935310247}{40682413003359742540366026580} a^{8} + \frac{6122441332180372104547647717}{20341206501679871270183013290} a^{7} + \frac{3383304918359345259360734551}{10170603250839935635091506645} a^{6} - \frac{777542368247539955039419969}{40682413003359742540366026580} a^{5} + \frac{3455601902522288635350901359}{40682413003359742540366026580} a^{4} + \frac{268202804133592189690915524}{10170603250839935635091506645} a^{3} + \frac{2631640177263466295915024417}{20341206501679871270183013290} a^{2} - \frac{513333149722624080833927244}{2034120650167987127018301329} a - \frac{25877207197357713235590982}{56191178181436108481168545}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{18}\times C_{18}$, which has order $2592$ (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:  \( 205236.825908 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-13}) \), 3.3.8281.1, 3.3.8281.2, 3.3.169.1, \(\Q(\zeta_{7})^+\), 6.0.57054367552.1, 6.0.57054367552.2, 6.0.23762752.1, 6.0.337599808.1, 9.9.567869252041.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$