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Properties

Label 18.0.21005238210...5344.6
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 7^{15}\cdot 19^{12}$
Root discriminant $374.51$
Ramified primes $2, 3, 7, 19$
Class number $112783104$ (GRH)
Class group $[2, 2, 2, 6, 36, 65268]$ (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![13691196826273, 6221781695196, 8571032362377, 1713810637074, 896870114343, 28166415822, -22592017839, -5860535940, -1724576913, 204598042, 105501465, -3430944, -2613396, 30096, 36072, -114, -279, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 279*x^16 - 114*x^15 + 36072*x^14 + 30096*x^13 - 2613396*x^12 - 3430944*x^11 + 105501465*x^10 + 204598042*x^9 - 1724576913*x^8 - 5860535940*x^7 - 22592017839*x^6 + 28166415822*x^5 + 896870114343*x^4 + 1713810637074*x^3 + 8571032362377*x^2 + 6221781695196*x + 13691196826273)
 
gp: K = bnfinit(x^18 - 279*x^16 - 114*x^15 + 36072*x^14 + 30096*x^13 - 2613396*x^12 - 3430944*x^11 + 105501465*x^10 + 204598042*x^9 - 1724576913*x^8 - 5860535940*x^7 - 22592017839*x^6 + 28166415822*x^5 + 896870114343*x^4 + 1713810637074*x^3 + 8571032362377*x^2 + 6221781695196*x + 13691196826273, 1)
 

Normalized defining polynomial

\( x^{18} - 279 x^{16} - 114 x^{15} + 36072 x^{14} + 30096 x^{13} - 2613396 x^{12} - 3430944 x^{11} + 105501465 x^{10} + 204598042 x^{9} - 1724576913 x^{8} - 5860535940 x^{7} - 22592017839 x^{6} + 28166415822 x^{5} + 896870114343 x^{4} + 1713810637074 x^{3} + 8571032362377 x^{2} + 6221781695196 x + 13691196826273 \)

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:  \(-21005238210476463669563930923405113491534905344=-\,2^{18}\cdot 3^{27}\cdot 7^{15}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $374.51$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(1,·)$, $\chi_{4788}(2053,·)$, $\chi_{4788}(961,·)$, $\chi_{4788}(2063,·)$, $\chi_{4788}(4115,·)$, $\chi_{4788}(277,·)$, $\chi_{4788}(1369,·)$, $\chi_{4788}(4751,·)$, $\chi_{4788}(2015,·)$, $\chi_{4788}(4225,·)$, $\chi_{4788}(2857,·)$, $\chi_{4788}(647,·)$, $\chi_{4788}(3697,·)$, $\chi_{4788}(311,·)$, $\chi_{4788}(121,·)$, $\chi_{4788}(2747,·)$, $\chi_{4788}(1679,·)$, $\chi_{4788}(4415,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{27} a^{6} + \frac{1}{9} a^{4} - \frac{2}{27} a^{3} + \frac{1}{3} a^{2} - \frac{1}{9} a + \frac{1}{27}$, $\frac{1}{27} a^{7} + \frac{4}{27} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} + \frac{4}{27} a - \frac{2}{9}$, $\frac{1}{54} a^{8} - \frac{1}{27} a^{5} - \frac{1}{6} a^{4} - \frac{1}{9} a^{3} + \frac{1}{54} a^{2} + \frac{1}{3} a - \frac{7}{18}$, $\frac{1}{162} a^{9} - \frac{1}{18} a^{5} + \frac{5}{54} a^{3} - \frac{1}{9} a^{2} + \frac{1}{6} a - \frac{8}{81}$, $\frac{1}{162} a^{10} - \frac{1}{54} a^{6} - \frac{7}{54} a^{4} + \frac{4}{27} a^{3} - \frac{1}{2} a^{2} + \frac{10}{81} a - \frac{8}{27}$, $\frac{1}{486} a^{11} + \frac{1}{486} a^{10} + \frac{1}{486} a^{9} - \frac{1}{162} a^{8} - \frac{1}{162} a^{7} - \frac{1}{162} a^{6} - \frac{4}{81} a^{5} - \frac{4}{81} a^{4} - \frac{4}{81} a^{3} + \frac{121}{243} a^{2} - \frac{1}{486} a - \frac{1}{486}$, $\frac{1}{7290} a^{12} - \frac{1}{2430} a^{11} - \frac{1}{1215} a^{10} + \frac{2}{729} a^{9} + \frac{1}{810} a^{8} + \frac{11}{810} a^{7} - \frac{13}{2430} a^{6} - \frac{37}{810} a^{5} - \frac{83}{810} a^{4} + \frac{1121}{7290} a^{3} - \frac{13}{1215} a^{2} - \frac{512}{1215} a - \frac{997}{3645}$, $\frac{1}{7290} a^{13} + \frac{17}{7290} a^{10} - \frac{1}{1215} a^{9} - \frac{1}{135} a^{8} - \frac{19}{2430} a^{7} + \frac{1}{162} a^{6} - \frac{4}{135} a^{5} + \frac{217}{1458} a^{4} + \frac{11}{162} a^{3} + \frac{37}{270} a^{2} - \frac{31}{1458} a - \frac{62}{1215}$, $\frac{1}{7290} a^{14} + \frac{1}{3645} a^{11} - \frac{7}{2430} a^{10} + \frac{7}{2430} a^{9} - \frac{2}{1215} a^{8} + \frac{1}{81} a^{7} + \frac{11}{810} a^{6} - \frac{35}{1458} a^{5} + \frac{19}{162} a^{4} - \frac{119}{810} a^{3} + \frac{53}{1458} a^{2} - \frac{929}{2430} a - \frac{185}{486}$, $\frac{1}{21870} a^{15} + \frac{1}{21870} a^{12} - \frac{1}{1215} a^{11} + \frac{1}{810} a^{10} + \frac{29}{10935} a^{9} - \frac{7}{810} a^{8} - \frac{68}{10935} a^{6} + \frac{17}{405} a^{5} - \frac{2}{135} a^{4} + \frac{1057}{10935} a^{3} + \frac{299}{2430} a^{2} + \frac{11}{810} a - \frac{533}{10935}$, $\frac{1}{4429070364490537750872608481660} a^{16} + \frac{810941239456475360969563}{246059464693918763937367137870} a^{15} - \frac{1972407783043925410333921}{123029732346959381968683568935} a^{14} + \frac{138823559200525667260667327}{2214535182245268875436304240830} a^{13} + \frac{2431916751706518018521209}{123029732346959381968683568935} a^{12} + \frac{35857463031388312740109994}{123029732346959381968683568935} a^{11} - \frac{804228756451242352074582923}{442907036449053775087260848166} a^{10} - \frac{209261037577862106254919401}{123029732346959381968683568935} a^{9} - \frac{51346309401826146394800071}{6075542338121450961416472540} a^{8} - \frac{7074172233107669841660713395}{442907036449053775087260848166} a^{7} - \frac{2127456178195424196806808857}{123029732346959381968683568935} a^{6} + \frac{2015651718087152507218180921}{123029732346959381968683568935} a^{5} - \frac{424475360211316699162480694983}{4429070364490537750872608481660} a^{4} - \frac{6239470366723806057264763523}{246059464693918763937367137870} a^{3} + \frac{107847069655425654356450617613}{246059464693918763937367137870} a^{2} + \frac{273011966993316377012130009437}{1107267591122634437718152120415} a - \frac{157714822984145563215541089883}{492118929387837527874734275740}$, $\frac{1}{3788834139190652719041176467730666168472440940} a^{17} + \frac{322040421344587}{3788834139190652719041176467730666168472440940} a^{16} - \frac{5822876226067235109764062164475705271456}{947208534797663179760294116932666542118110235} a^{15} + \frac{91806328310859198161556462681112534106483}{1894417069595326359520588233865333084236220470} a^{14} - \frac{24313217892222823682003625323651564455832}{947208534797663179760294116932666542118110235} a^{13} - \frac{890461194257051726286896726945396923462}{23102647190186906823421807730065037612636835} a^{12} + \frac{1299117361995833114173399388177075552643947}{1894417069595326359520588233865333084236220470} a^{11} + \frac{497305464666841939886289899623146385112774}{189441706959532635952058823386533308423622047} a^{10} - \frac{1418039589679095526939781131675661431607933}{3788834139190652719041176467730666168472440940} a^{9} + \frac{9976430842205230556843132384903424393489023}{3788834139190652719041176467730666168472440940} a^{8} + \frac{1443472882791056702595970519590161189261539}{189441706959532635952058823386533308423622047} a^{7} - \frac{14402934526498245653665868122960276786259687}{1894417069595326359520588233865333084236220470} a^{6} + \frac{26432257243194632390395858823491061366341631}{3788834139190652719041176467730666168472440940} a^{5} - \frac{59891663363534850039116622679013515207257287}{757766827838130543808235293546133233694488188} a^{4} - \frac{11437241515550613475174050865543113760765888}{947208534797663179760294116932666542118110235} a^{3} - \frac{224623581708957047573538371536563157374041146}{947208534797663179760294116932666542118110235} a^{2} + \frac{57441501705283562498674874628494651970399595}{757766827838130543808235293546133233694488188} a + \frac{401627643295030761416530195291480268971785353}{3788834139190652719041176467730666168472440940}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{36}\times C_{65268}$, which has order $112783104$ (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:  \( 42198260.232521206 \) (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

\(\Q(\sqrt{-21}) \), 3.3.1432809.2, 3.3.29241.2, \(\Q(\zeta_{7})^+\), 3.3.1432809.1, 6.0.2759153551366464.5, 6.0.56309256150336.2, 6.0.29042496.1, 6.0.2759153551366464.4, 9.9.2941473244627851129.4

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 R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.5$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$$[2]^{3}$
3Data not computed
7Data not computed
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$