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Properties

Label 18.0.10561389012...0663.9
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{15}\cdot 31^{12}$
Root discriminant $216.10$
Ramified primes $3, 7, 31$
Class number $15926169$ (GRH)
Class group $[15926169]$ (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![921751552, 498077184, -445115136, -8290208, 88036464, -49153200, 5953192, -741978, 2166711, -1108894, 216501, 21564, 3702, -8460, 1074, 286, -45, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 45*x^16 + 286*x^15 + 1074*x^14 - 8460*x^13 + 3702*x^12 + 21564*x^11 + 216501*x^10 - 1108894*x^9 + 2166711*x^8 - 741978*x^7 + 5953192*x^6 - 49153200*x^5 + 88036464*x^4 - 8290208*x^3 - 445115136*x^2 + 498077184*x + 921751552)
 
gp: K = bnfinit(x^18 - 6*x^17 - 45*x^16 + 286*x^15 + 1074*x^14 - 8460*x^13 + 3702*x^12 + 21564*x^11 + 216501*x^10 - 1108894*x^9 + 2166711*x^8 - 741978*x^7 + 5953192*x^6 - 49153200*x^5 + 88036464*x^4 - 8290208*x^3 - 445115136*x^2 + 498077184*x + 921751552, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 45 x^{16} + 286 x^{15} + 1074 x^{14} - 8460 x^{13} + 3702 x^{12} + 21564 x^{11} + 216501 x^{10} - 1108894 x^{9} + 2166711 x^{8} - 741978 x^{7} + 5953192 x^{6} - 49153200 x^{5} + 88036464 x^{4} - 8290208 x^{3} - 445115136 x^{2} + 498077184 x + 921751552 \)

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:  \(-1056138901279026047519828726794817132040663=-\,3^{24}\cdot 7^{15}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $216.10$
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:  $3, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1953=3^{2}\cdot 7\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1953}(1,·)$, $\chi_{1953}(1741,·)$, $\chi_{1953}(718,·)$, $\chi_{1953}(1489,·)$, $\chi_{1953}(466,·)$, $\chi_{1953}(1048,·)$, $\chi_{1953}(25,·)$, $\chi_{1953}(1885,·)$, $\chi_{1953}(997,·)$, $\chi_{1953}(1513,·)$, $\chi_{1953}(811,·)$, $\chi_{1953}(559,·)$, $\chi_{1953}(304,·)$, $\chi_{1953}(625,·)$, $\chi_{1953}(373,·)$, $\chi_{1953}(118,·)$, $\chi_{1953}(745,·)$, $\chi_{1953}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{256} a^{9} - \frac{1}{256} a^{8} - \frac{1}{128} a^{7} - \frac{1}{128} a^{6} + \frac{9}{256} a^{5} + \frac{15}{256} a^{4} + \frac{1}{32} a^{3} - \frac{3}{64} a^{2} + \frac{7}{16} a - \frac{1}{2}$, $\frac{1}{512} a^{10} - \frac{1}{512} a^{9} - \frac{1}{256} a^{8} - \frac{1}{256} a^{7} + \frac{9}{512} a^{6} + \frac{15}{512} a^{5} + \frac{1}{64} a^{4} + \frac{29}{128} a^{3} - \frac{1}{32} a^{2} - \frac{1}{4} a$, $\frac{1}{512} a^{11} - \frac{1}{512} a^{9} - \frac{1}{256} a^{8} - \frac{13}{512} a^{7} - \frac{1}{128} a^{6} - \frac{23}{512} a^{5} - \frac{1}{256} a^{4} - \frac{31}{128} a^{3} + \frac{1}{64} a^{2} - \frac{3}{16} a - \frac{1}{2}$, $\frac{1}{2048} a^{12} - \frac{1}{1024} a^{11} + \frac{1}{2048} a^{10} - \frac{1}{1024} a^{9} + \frac{7}{2048} a^{8} - \frac{31}{1024} a^{7} - \frac{53}{2048} a^{6} - \frac{27}{1024} a^{5} - \frac{13}{512} a^{4} + \frac{47}{256} a^{3} - \frac{13}{64} a^{2} - \frac{3}{8} a$, $\frac{1}{4096} a^{13} - \frac{3}{4096} a^{11} + \frac{3}{4096} a^{9} - \frac{113}{4096} a^{7} + \frac{1}{64} a^{6} - \frac{5}{128} a^{5} - \frac{1}{64} a^{4} - \frac{31}{256} a^{3} - \frac{1}{8} a^{2} - \frac{7}{16} a - \frac{1}{2}$, $\frac{1}{8192} a^{14} - \frac{1}{8192} a^{13} + \frac{1}{8192} a^{12} + \frac{3}{8192} a^{11} + \frac{7}{8192} a^{10} - \frac{3}{8192} a^{9} - \frac{21}{8192} a^{8} - \frac{79}{8192} a^{7} - \frac{13}{2048} a^{6} - \frac{5}{256} a^{5} + \frac{7}{128} a^{4} - \frac{1}{512} a^{3} - \frac{3}{64} a^{2} - \frac{7}{32} a - \frac{1}{4}$, $\frac{1}{65536} a^{15} - \frac{1}{16384} a^{14} + \frac{3}{16384} a^{12} - \frac{31}{32768} a^{11} - \frac{3}{16384} a^{10} - \frac{3}{2048} a^{9} - \frac{63}{16384} a^{8} + \frac{1221}{65536} a^{7} + \frac{3}{128} a^{6} - \frac{111}{8192} a^{5} - \frac{7}{128} a^{4} - \frac{987}{4096} a^{3} - \frac{31}{256} a^{2} + \frac{45}{256} a + \frac{3}{32}$, $\frac{1}{65536} a^{16} - \frac{1}{16384} a^{13} + \frac{1}{32768} a^{12} + \frac{11}{16384} a^{11} - \frac{1}{2048} a^{10} + \frac{21}{16384} a^{9} - \frac{123}{65536} a^{8} + \frac{457}{16384} a^{7} + \frac{41}{8192} a^{6} + \frac{57}{2048} a^{5} - \frac{155}{4096} a^{4} - \frac{235}{1024} a^{3} - \frac{39}{256} a^{2} - \frac{9}{64} a - \frac{1}{8}$, $\frac{1}{404514815588405969675836620891269103616} a^{17} - \frac{2177755324376312533936964523336855}{404514815588405969675836620891269103616} a^{16} + \frac{27793664791573796404299999637253}{202257407794202984837918310445634551808} a^{15} - \frac{5463504670694349104332284199707615}{101128703897101492418959155222817275904} a^{14} + \frac{18488309791114447478031707173369751}{202257407794202984837918310445634551808} a^{13} - \frac{16694522964169908054677066171181421}{202257407794202984837918310445634551808} a^{12} + \frac{15404455592538062045885961874941889}{25282175974275373104739788805704318976} a^{11} + \frac{9094437019263990932703356404891235}{101128703897101492418959155222817275904} a^{10} - \frac{363236570618669173527057376786065239}{404514815588405969675836620891269103616} a^{9} - \frac{471701598971247325946868600080539031}{404514815588405969675836620891269103616} a^{8} + \frac{5926156553123937624358138063660380663}{202257407794202984837918310445634551808} a^{7} + \frac{219818262789685510033567183320329917}{50564351948550746209479577611408637952} a^{6} - \frac{188506293317007403270984893035419489}{6320543993568843276184947201426079744} a^{5} - \frac{1378226358857315503136446157708073247}{25282175974275373104739788805704318976} a^{4} - \frac{1381100745611837800239351297980272349}{12641087987137686552369894402852159488} a^{3} - \frac{373697279191021675184718397777922377}{1580135998392210819046236800356519936} a^{2} - \frac{44974079123218532047335753287037321}{790067999196105409523118400178259968} a + \frac{2911319116861135482551538084900597}{98758499899513176190389800022282496}$

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

Class group and class number

$C_{15926169}$, which has order $15926169$ (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:  \( 219311185150.01114 \) (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

\(\Q(\sqrt{-7}) \), 3.3.3814209.2, 3.3.3814209.1, 3.3.3969.1, 3.3.961.1, 6.0.101837332069767.2, 6.0.101837332069767.1, 6.0.110270727.2, 6.0.316767703.1, 9.9.55489838359499131329.9

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
7Data not computed
$31$31.6.4.1$x^{6} + 1085 x^{3} + 1660608$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.6.4.1$x^{6} + 1085 x^{3} + 1660608$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.6.4.1$x^{6} + 1085 x^{3} + 1660608$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$