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Properties

Label 18.0.54960223726...4559.5
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{15}\cdot 19^{15}$
Root discriminant $305.89$
Ramified primes $3, 7, 19$
Class number $8893108992$ (GRH)
Class group $[2, 2, 2, 2, 2, 277909656]$ (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![2772434944000, 2249313484800, 1332058521600, 424503642112, 110919615744, 12409534080, 1659031088, -291814320, 44404560, -12808812, 3932949, -393084, 216480, -6276, 5202, -100, 120, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 120*x^16 - 100*x^15 + 5202*x^14 - 6276*x^13 + 216480*x^12 - 393084*x^11 + 3932949*x^10 - 12808812*x^9 + 44404560*x^8 - 291814320*x^7 + 1659031088*x^6 + 12409534080*x^5 + 110919615744*x^4 + 424503642112*x^3 + 1332058521600*x^2 + 2249313484800*x + 2772434944000)
 
gp: K = bnfinit(x^18 + 120*x^16 - 100*x^15 + 5202*x^14 - 6276*x^13 + 216480*x^12 - 393084*x^11 + 3932949*x^10 - 12808812*x^9 + 44404560*x^8 - 291814320*x^7 + 1659031088*x^6 + 12409534080*x^5 + 110919615744*x^4 + 424503642112*x^3 + 1332058521600*x^2 + 2249313484800*x + 2772434944000, 1)
 

Normalized defining polynomial

\( x^{18} + 120 x^{16} - 100 x^{15} + 5202 x^{14} - 6276 x^{13} + 216480 x^{12} - 393084 x^{11} + 3932949 x^{10} - 12808812 x^{9} + 44404560 x^{8} - 291814320 x^{7} + 1659031088 x^{6} + 12409534080 x^{5} + 110919615744 x^{4} + 424503642112 x^{3} + 1332058521600 x^{2} + 2249313484800 x + 2772434944000 \)

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:  \(-549602237265236146200328835310499852899314559=-\,3^{27}\cdot 7^{15}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $305.89$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1197=3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1197}(1,·)$, $\chi_{1197}(961,·)$, $\chi_{1197}(904,·)$, $\chi_{1197}(521,·)$, $\chi_{1197}(970,·)$, $\chi_{1197}(335,·)$, $\chi_{1197}(563,·)$, $\chi_{1197}(278,·)$, $\chi_{1197}(919,·)$, $\chi_{1197}(634,·)$, $\chi_{1197}(862,·)$, $\chi_{1197}(227,·)$, $\chi_{1197}(676,·)$, $\chi_{1197}(293,·)$, $\chi_{1197}(1196,·)$, $\chi_{1197}(1139,·)$, $\chi_{1197}(236,·)$, $\chi_{1197}(58,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{32} a^{7} - \frac{1}{32} a^{6} - \frac{1}{32} a^{5} + \frac{1}{32} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{64} a^{6} + \frac{1}{32} a^{5} + \frac{5}{128} a^{4} - \frac{5}{32} a^{3} + \frac{7}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{128} a^{9} - \frac{1}{64} a^{7} - \frac{1}{32} a^{6} - \frac{3}{128} a^{5} + \frac{1}{32} a^{4} - \frac{7}{32} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{256} a^{10} - \frac{1}{256} a^{9} - \frac{1}{128} a^{7} + \frac{5}{256} a^{6} - \frac{9}{256} a^{5} - \frac{7}{128} a^{4} + \frac{11}{64} a^{3} - \frac{7}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{1024} a^{11} + \frac{1}{1024} a^{10} + \frac{1}{512} a^{8} + \frac{5}{1024} a^{7} - \frac{7}{1024} a^{6} + \frac{9}{512} a^{5} - \frac{7}{256} a^{4} + \frac{5}{128} a^{3} - \frac{1}{32} a^{2}$, $\frac{1}{1024} a^{12} - \frac{1}{1024} a^{10} + \frac{1}{512} a^{9} + \frac{3}{1024} a^{8} - \frac{3}{256} a^{7} + \frac{25}{1024} a^{6} - \frac{23}{512} a^{5} - \frac{15}{256} a^{4} + \frac{23}{128} a^{3} + \frac{5}{32} a^{2} - \frac{1}{4} a$, $\frac{1}{20480} a^{13} - \frac{1}{2048} a^{12} - \frac{7}{20480} a^{11} - \frac{13}{10240} a^{10} - \frac{49}{20480} a^{9} - \frac{11}{10240} a^{8} + \frac{179}{20480} a^{7} + \frac{17}{10240} a^{6} - \frac{63}{5120} a^{5} - \frac{93}{2560} a^{4} - \frac{7}{32} a^{3} - \frac{1}{16} a^{2} - \frac{19}{40} a$, $\frac{1}{81920} a^{14} + \frac{1}{81920} a^{13} - \frac{37}{81920} a^{12} + \frac{17}{81920} a^{11} - \frac{11}{16384} a^{10} - \frac{161}{81920} a^{9} - \frac{223}{81920} a^{8} - \frac{437}{81920} a^{7} - \frac{679}{40960} a^{6} - \frac{399}{20480} a^{5} + \frac{417}{10240} a^{4} + \frac{9}{128} a^{3} + \frac{159}{640} a^{2} + \frac{23}{80} a$, $\frac{1}{327680} a^{15} + \frac{1}{327680} a^{14} - \frac{1}{327680} a^{13} - \frac{103}{327680} a^{12} + \frac{93}{327680} a^{11} + \frac{23}{327680} a^{10} - \frac{547}{327680} a^{9} - \frac{989}{327680} a^{8} + \frac{503}{163840} a^{7} - \frac{1293}{81920} a^{6} - \frac{757}{40960} a^{5} - \frac{637}{10240} a^{4} - \frac{81}{2560} a^{3} + \frac{91}{640} a^{2} - \frac{13}{80} a$, $\frac{1}{327680} a^{16} - \frac{1}{163840} a^{14} - \frac{3}{163840} a^{13} - \frac{31}{81920} a^{12} - \frac{51}{163840} a^{11} - \frac{253}{163840} a^{10} + \frac{627}{163840} a^{9} + \frac{523}{327680} a^{8} - \frac{1857}{163840} a^{7} - \frac{457}{16384} a^{6} - \frac{3}{8192} a^{5} - \frac{79}{10240} a^{4} + \frac{9}{512} a^{3} + \frac{5}{128} a^{2} + \frac{3}{16} a$, $\frac{1}{2172792705126322646425426626318628388264387799795278754611200} a^{17} - \frac{24003728764988335215536937097421684508685233731844693}{27159908814079033080317832828982854853304847497440984432640} a^{16} - \frac{12650847933742010842046630114832023696367451475181417}{54319817628158066160635665657965709706609694994881968865280} a^{15} - \frac{37774529686586619861693472554404210599028186367032009}{21727927051263226464254266263186283882643877997952787546112} a^{14} - \frac{26093481316090580193900714059373052176284888433347699879}{1086396352563161323212713313159314194132193899897639377305600} a^{13} - \frac{156953945744642283427868774972754048344141321869466895089}{543198176281580661606356656579657097066096949948819688652800} a^{12} + \frac{1881014610572403952140448945662040096419194282866257347}{6789977203519758270079458207245713713326211874360246108160} a^{11} - \frac{1027329004632104826422118115634835166619246338076469145431}{543198176281580661606356656579657097066096949948819688652800} a^{10} + \frac{4421219381461514910530377088578753159943879931996121616949}{2172792705126322646425426626318628388264387799795278754611200} a^{9} + \frac{1838634317573429913753813647854259756011728482245673336537}{543198176281580661606356656579657097066096949948819688652800} a^{8} + \frac{56459216246373615331618684027928170920574095007158539265}{5431981762815806616063566565796570970660969499488196886528} a^{7} - \frac{392717768536071855547775714879770577929710075898351652847}{27159908814079033080317832828982854853304847497440984432640} a^{6} - \frac{269605229402277786299294802063154306227834553955181656237}{135799544070395165401589164144914274266524237487204922163200} a^{5} + \frac{3752495165804880267864815640581786159701434950923215915}{678997720351975827007945820724571371332621187436024610816} a^{4} + \frac{763331489063322256736523201906557370412099589277491399549}{8487471504399697837599322759057142141657764842950307635200} a^{3} + \frac{180068892777162206390949697881395216202418029738218698699}{1060933938049962229699915344882142767707220605368788454400} a^{2} + \frac{1290660167705657682109060902016594982941272880299862387}{2652334845124905574249788362205356919268051513421971136} a + \frac{10965542985752499443513881596936688113111870023926380}{41442731955076649597652943159458701863563304897218299}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{277909656}$, which has order $8893108992$ (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:  \( 10681224266.072006 \) (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

\(\Q(\sqrt{-399}) \), 3.3.29241.1, 3.3.1432809.2, 3.3.17689.2, 3.3.3969.1, 6.0.16716810419631.3, 6.0.819123710561919.3, 6.0.1123626489111.2, 6.0.2269040749479.8, 9.9.2941473244627851129.8

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
19Data not computed