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Properties

Label 18.0.88267617078...823.10
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{15}\cdot 37^{12}$
Root discriminant $243.15$
Ramified primes $3, 7, 37$
Class number $18729711$ (GRH)
Class group $[3, 693, 9009]$ (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![25516048384, 168742637568, 389859017472, 298107348480, 109788683088, 11462360976, -6893038592, -2622648420, -29319813, 145388304, 18110517, -3918456, -814130, 54144, 17658, -324, -201, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 201*x^16 - 324*x^15 + 17658*x^14 + 54144*x^13 - 814130*x^12 - 3918456*x^11 + 18110517*x^10 + 145388304*x^9 - 29319813*x^8 - 2622648420*x^7 - 6893038592*x^6 + 11462360976*x^5 + 109788683088*x^4 + 298107348480*x^3 + 389859017472*x^2 + 168742637568*x + 25516048384)
 
gp: K = bnfinit(x^18 - 201*x^16 - 324*x^15 + 17658*x^14 + 54144*x^13 - 814130*x^12 - 3918456*x^11 + 18110517*x^10 + 145388304*x^9 - 29319813*x^8 - 2622648420*x^7 - 6893038592*x^6 + 11462360976*x^5 + 109788683088*x^4 + 298107348480*x^3 + 389859017472*x^2 + 168742637568*x + 25516048384, 1)
 

Normalized defining polynomial

\( x^{18} - 201 x^{16} - 324 x^{15} + 17658 x^{14} + 54144 x^{13} - 814130 x^{12} - 3918456 x^{11} + 18110517 x^{10} + 145388304 x^{9} - 29319813 x^{8} - 2622648420 x^{7} - 6893038592 x^{6} + 11462360976 x^{5} + 109788683088 x^{4} + 298107348480 x^{3} + 389859017472 x^{2} + 168742637568 x + 25516048384 \)

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:  \(-8826761707820991064360908575466167817698823=-\,3^{24}\cdot 7^{15}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $243.15$
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:  $3, 7, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2331=3^{2}\cdot 7\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{2331}(1,·)$, $\chi_{2331}(898,·)$, $\chi_{2331}(1987,·)$, $\chi_{2331}(454,·)$, $\chi_{2331}(1543,·)$, $\chi_{2331}(10,·)$, $\chi_{2331}(1111,·)$, $\chi_{2331}(1444,·)$, $\chi_{2331}(988,·)$, $\chi_{2331}(100,·)$, $\chi_{2331}(2209,·)$, $\chi_{2331}(676,·)$, $\chi_{2331}(1222,·)$, $\chi_{2331}(1000,·)$, $\chi_{2331}(556,·)$, $\chi_{2331}(2098,·)$, $\chi_{2331}(565,·)$, $\chi_{2331}(1786,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{7} + \frac{1}{64} a^{6} + \frac{3}{64} a^{5} + \frac{3}{64} a^{4} - \frac{3}{16} a^{3} - \frac{3}{16} a^{2}$, $\frac{1}{128} a^{8} - \frac{1}{64} a^{6} - \frac{1}{32} a^{5} + \frac{5}{128} a^{4} + \frac{5}{32} a^{3} + \frac{7}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{9} - \frac{1}{64} a^{6} - \frac{5}{128} a^{5} - \frac{3}{64} a^{4} + \frac{5}{32} a^{3} + \frac{3}{16} a^{2}$, $\frac{1}{512} a^{10} + \frac{1}{512} a^{9} - \frac{1}{256} a^{8} + \frac{1}{256} a^{7} - \frac{7}{512} a^{6} + \frac{1}{512} a^{5} - \frac{1}{64} a^{4} + \frac{31}{128} a^{3} - \frac{7}{32} a^{2} - \frac{1}{2} a$, $\frac{1}{2048} a^{11} - \frac{1}{2048} a^{10} + \frac{1}{512} a^{9} + \frac{3}{1024} a^{8} + \frac{13}{2048} a^{7} + \frac{55}{2048} a^{6} + \frac{59}{1024} a^{5} - \frac{31}{512} a^{4} + \frac{39}{256} a^{3} + \frac{3}{16} a^{2} - \frac{1}{16} a$, $\frac{1}{10240} a^{12} - \frac{1}{5120} a^{11} + \frac{1}{10240} a^{10} - \frac{1}{5120} a^{9} + \frac{3}{2048} a^{8} - \frac{3}{1024} a^{7} + \frac{31}{2048} a^{6} + \frac{229}{5120} a^{5} - \frac{91}{2560} a^{4} + \frac{251}{1280} a^{3} - \frac{29}{160} a^{2} - \frac{3}{80} a - \frac{2}{5}$, $\frac{1}{40960} a^{13} + \frac{7}{40960} a^{11} - \frac{1}{4096} a^{10} + \frac{51}{40960} a^{9} + \frac{3}{2048} a^{8} + \frac{13}{8192} a^{7} + \frac{259}{20480} a^{6} - \frac{487}{10240} a^{5} + \frac{5}{1024} a^{4} + \frac{49}{256} a^{3} + \frac{53}{320} a^{2} + \frac{3}{80} a - \frac{1}{5}$, $\frac{1}{81920} a^{14} - \frac{1}{81920} a^{13} + \frac{3}{81920} a^{12} - \frac{9}{81920} a^{11} + \frac{57}{81920} a^{10} - \frac{303}{81920} a^{9} + \frac{53}{16384} a^{8} + \frac{573}{81920} a^{7} - \frac{263}{40960} a^{6} - \frac{481}{20480} a^{5} - \frac{23}{10240} a^{4} - \frac{131}{640} a^{3} - \frac{53}{640} a^{2} + \frac{27}{80} a - \frac{1}{5}$, $\frac{1}{120750080} a^{15} - \frac{1}{1097728} a^{14} - \frac{73}{15093760} a^{13} + \frac{153}{15093760} a^{12} + \frac{963}{60375040} a^{11} - \frac{2109}{2744320} a^{10} + \frac{107829}{30187520} a^{9} - \frac{23847}{7546880} a^{8} + \frac{168737}{120750080} a^{7} - \frac{1092499}{60375040} a^{6} - \frac{40505}{6037504} a^{5} + \frac{309031}{15093760} a^{4} - \frac{155043}{1886720} a^{3} - \frac{2549}{17152} a^{2} + \frac{1863}{29480} a + \frac{861}{3685}$, $\frac{1}{88872058880} a^{16} - \frac{1}{386400256} a^{15} - \frac{131511}{44436029440} a^{14} - \frac{271677}{44436029440} a^{13} - \frac{86723}{22218014720} a^{12} - \frac{4608279}{44436029440} a^{11} - \frac{3970777}{44436029440} a^{10} - \frac{10460293}{4039639040} a^{9} - \frac{306264693}{88872058880} a^{8} + \frac{121564307}{22218014720} a^{7} + \frac{6833161}{1110900736} a^{6} + \frac{346430571}{5554503680} a^{5} - \frac{307808819}{5554503680} a^{4} + \frac{20720663}{86789120} a^{3} + \frac{10955487}{69431296} a^{2} + \frac{6219549}{21697280} a - \frac{1334}{3685}$, $\frac{1}{476599205105637369336260004228451315283394560} a^{17} + \frac{62494544107054201266409022209}{1883791324528210945993122546357515080171520} a^{16} - \frac{1352972358744728221742855520041923}{7446862579775583895879062566069551801303040} a^{15} - \frac{221273525810061754682193848666192282561}{119149801276409342334065001057112828820848640} a^{14} - \frac{1869149272631907531459685910531413849853}{238299602552818684668130002114225657641697280} a^{13} - \frac{4840832350841248818758212319978359982409}{238299602552818684668130002114225657641697280} a^{12} - \frac{18671316168779122734662559767706072971551}{119149801276409342334065001057112828820848640} a^{11} - \frac{29197843951586925258780678756860071309121}{119149801276409342334065001057112828820848640} a^{10} - \frac{10923311441703850742241569586768723467933}{43327200464148851757841818566222846843944960} a^{9} + \frac{1393239754304953210836811297667112040398157}{476599205105637369336260004228451315283394560} a^{8} + \frac{477116764693110656385218087535905521733477}{119149801276409342334065001057112828820848640} a^{7} - \frac{437818353103583082689313479101050915640137}{14893725159551167791758125132139103602606080} a^{6} + \frac{505451318577720135428109115528538808505467}{14893725159551167791758125132139103602606080} a^{5} - \frac{329864286492716020565500083662417980445845}{5957490063820467116703250052855641441042432} a^{4} + \frac{109707076668891346900044088868270713615143}{1861715644943895973969765641517387950325760} a^{3} + \frac{423479243256908691162366053794180816673713}{1861715644943895973969765641517387950325760} a^{2} + \frac{945886370965091701185661958329076015681}{5059009904738847755352624025862467256320} a - \frac{175622023634523064443951472631142051}{429603422617089653137960600022288320}$

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

Class group and class number

$C_{3}\times C_{693}\times C_{9009}$, which has order $18729711$ (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:  \( 958454033532.8324 \) (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

\(\Q(\sqrt{-7}) \), 3.3.110889.2, 3.3.67081.1, 3.3.3969.1, 3.3.5433561.1, 6.0.4217655020103.6, 6.0.31499023927.2, 6.0.110270727.2, 6.0.206665095985047.2, 9.9.160418200800801137481.10

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.8.6$x^{6} + 18 x^{2} + 36$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.6$x^{6} + 18 x^{2} + 36$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.6$x^{6} + 18 x^{2} + 36$$3$$2$$8$$C_6$$[2]^{2}$
7Data not computed
$37$37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$