↑

Properties

Label 18.0.17582592502...0128.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{53}\cdot 7^{12}$
Root discriminant $172.12$
Ramified primes $2, 3, 7$
Class number $243$ (GRH)
Class group $[3, 3, 3, 3, 3]$ (GRH)
Galois group $D_9:C_3$ (as 18T18)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3087691489, -500103, -2000331, 9334932, -7000902, -7002198, 9336642, -2002680, -497187, 107993, 2907, -2304, 1554, -882, 414, -156, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 414*x^14 - 882*x^13 + 1554*x^12 - 2304*x^11 + 2907*x^10 + 107993*x^9 - 497187*x^8 - 2002680*x^7 + 9336642*x^6 - 7002198*x^5 - 7000902*x^4 + 9334932*x^3 - 2000331*x^2 - 500103*x + 3087691489)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 414*x^14 - 882*x^13 + 1554*x^12 - 2304*x^11 + 2907*x^10 + 107993*x^9 - 497187*x^8 - 2002680*x^7 + 9336642*x^6 - 7002198*x^5 - 7000902*x^4 + 9334932*x^3 - 2000331*x^2 - 500103*x + 3087691489, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 414 x^{14} - 882 x^{13} + 1554 x^{12} - 2304 x^{11} + 2907 x^{10} + 107993 x^{9} - 497187 x^{8} - 2002680 x^{7} + 9336642 x^{6} - 7002198 x^{5} - 7000902 x^{4} + 9334932 x^{3} - 2000331 x^{2} - 500103 x + 3087691489 \)

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:  \(-17582592502916589732558131885417583280128=-\,2^{16}\cdot 3^{53}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $172.12$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{63} a^{6} - \frac{1}{21} a^{5} + \frac{2}{21} a^{4} - \frac{1}{9} a^{3} + \frac{2}{21} a^{2} - \frac{1}{21} a + \frac{1}{63}$, $\frac{1}{189} a^{7} + \frac{1}{189} a^{6} - \frac{2}{63} a^{5} + \frac{17}{189} a^{4} - \frac{22}{189} a^{3} + \frac{1}{9} a^{2} - \frac{11}{189} a + \frac{4}{189}$, $\frac{1}{189} a^{8} - \frac{1}{189} a^{6} + \frac{5}{189} a^{5} - \frac{1}{63} a^{4} + \frac{1}{189} a^{3} + \frac{4}{189} a^{2} - \frac{1}{63} a + \frac{2}{189}$, $\frac{1}{567} a^{9} + \frac{1}{189} a^{6} + \frac{1}{189} a^{3} + \frac{1}{567}$, $\frac{1}{567} a^{10} - \frac{1}{189} a^{6} + \frac{2}{63} a^{5} - \frac{16}{189} a^{4} + \frac{22}{189} a^{3} - \frac{1}{9} a^{2} + \frac{34}{567} a - \frac{4}{189}$, $\frac{1}{1701} a^{11} - \frac{1}{1701} a^{10} + \frac{1}{1701} a^{9} + \frac{1}{567} a^{8} - \frac{1}{567} a^{7} + \frac{1}{567} a^{6} + \frac{1}{567} a^{5} - \frac{1}{567} a^{4} + \frac{1}{567} a^{3} - \frac{566}{1701} a^{2} + \frac{566}{1701} a - \frac{566}{1701}$, $\frac{1}{11907} a^{12} + \frac{1}{11907} a^{11} - \frac{1}{1701} a^{10} - \frac{1}{11907} a^{9} - \frac{5}{3969} a^{8} - \frac{1}{567} a^{7} + \frac{4}{1323} a^{6} - \frac{8}{567} a^{5} + \frac{65}{3969} a^{4} - \frac{218}{11907} a^{3} - \frac{554}{1701} a^{2} + \frac{3935}{11907} a - \frac{3982}{11907}$, $\frac{1}{35721} a^{13} + \frac{1}{35721} a^{12} + \frac{13}{35721} a^{10} + \frac{13}{35721} a^{9} - \frac{16}{11907} a^{7} - \frac{10}{1701} a^{6} + \frac{4}{147} a^{5} + \frac{5620}{35721} a^{4} - \frac{146}{5103} a^{3} - \frac{11}{147} a^{2} + \frac{9325}{35721} a - \frac{635}{5103}$, $\frac{1}{35721} a^{14} - \frac{1}{35721} a^{12} - \frac{8}{35721} a^{11} + \frac{1}{1701} a^{10} + \frac{29}{35721} a^{9} + \frac{26}{11907} a^{8} + \frac{10}{3969} a^{7} + \frac{58}{11907} a^{6} - \frac{20}{5103} a^{5} - \frac{311}{3969} a^{4} - \frac{5494}{35721} a^{3} + \frac{283}{729} a^{2} - \frac{5654}{11907} a + \frac{1208}{5103}$, $\frac{1}{107163} a^{15} - \frac{4}{107163} a^{12} - \frac{2}{11907} a^{11} - \frac{1}{1701} a^{10} - \frac{17}{107163} a^{9} + \frac{4}{3969} a^{8} - \frac{1}{567} a^{7} - \frac{746}{107163} a^{6} + \frac{13}{567} a^{5} + \frac{290}{3969} a^{4} + \frac{16034}{107163} a^{3} - \frac{44}{1701} a^{2} + \frac{1451}{11907} a + \frac{11665}{107163}$, $\frac{1}{107163} a^{16} - \frac{1}{107163} a^{13} + \frac{1}{35721} a^{12} + \frac{2}{11907} a^{11} + \frac{22}{107163} a^{10} + \frac{1}{35721} a^{9} - \frac{10}{3969} a^{8} + \frac{244}{107163} a^{7} - \frac{19}{11907} a^{6} - \frac{22}{3969} a^{5} + \frac{11834}{107163} a^{4} + \frac{4180}{35721} a^{3} - \frac{4}{243} a^{2} + \frac{13072}{107163} a + \frac{3751}{35721}$, $\frac{1}{74967642937758057651914104189013445} a^{17} - \frac{314577275424475136153860398868}{74967642937758057651914104189013445} a^{16} + \frac{332288150053505164107146516587}{74967642937758057651914104189013445} a^{15} + \frac{164520450537805148071838868221}{74967642937758057651914104189013445} a^{14} + \frac{200439676997080760830188674396}{14993528587551611530382820837802689} a^{13} + \frac{817556852984624263691816594618}{74967642937758057651914104189013445} a^{12} + \frac{5984537114737127735994288799252}{74967642937758057651914104189013445} a^{11} + \frac{38135079332638204015497323292383}{74967642937758057651914104189013445} a^{10} + \frac{9652843694400850823137074606257}{14993528587551611530382820837802689} a^{9} + \frac{63155283269944377030549791951053}{74967642937758057651914104189013445} a^{8} + \frac{19382509844803795612083884220518}{10709663276822579664559157741287635} a^{7} - \frac{422588591875628378421724274299124}{74967642937758057651914104189013445} a^{6} - \frac{2511626598180732669304479595297672}{74967642937758057651914104189013445} a^{5} - \frac{121344869133507729451415389150021}{14993528587551611530382820837802689} a^{4} + \frac{11111423137761148482646253025599303}{74967642937758057651914104189013445} a^{3} + \frac{482660566433364174565351752017135}{2141932655364515932911831548257527} a^{2} - \frac{22582752754570311271469866047371086}{74967642937758057651914104189013445} a - \frac{367230770895460713646951085027}{1349139650111722022997716345835}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}$, which has order $243$ (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:  \( -\frac{122818039733767343782402}{169994655187659994675542186369645} a^{17} + \frac{1029501168875058676123471}{169994655187659994675542186369645} a^{16} - \frac{1787638265196755788713328}{56664885062553331558514062123215} a^{15} + \frac{19596959373813143802884398}{169994655187659994675542186369645} a^{14} - \frac{11687582566334219600912150}{33998931037531998935108437273929} a^{13} + \frac{48122343300105484671692788}{56664885062553331558514062123215} a^{12} - \frac{323117509657967471980186654}{169994655187659994675542186369645} a^{11} + \frac{303484707074629344984581314}{169994655187659994675542186369645} a^{10} + \frac{28562598771399899412516082}{11332977012510666311702812424643} a^{9} - \frac{19858186158758875250870871106}{169994655187659994675542186369645} a^{8} + \frac{72255137535627250828011125458}{169994655187659994675542186369645} a^{7} + \frac{53170115817179344059969572666}{56664885062553331558514062123215} a^{6} - \frac{696653633811359929163614951796}{169994655187659994675542186369645} a^{5} + \frac{169287679728737556378144790282}{33998931037531998935108437273929} a^{4} - \frac{146276919117251444841517300862}{56664885062553331558514062123215} a^{3} + \frac{303532079639653677429595742608}{33998931037531998935108437273929} a^{2} - \frac{20062950336911523073652885183948}{169994655187659994675542186369645} a + \frac{615096245646949568693857538}{1019757861006592609975598145} \) (order $6$)
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:  \( 157565908237.57254 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9:C_3$ (as 18T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $D_9:C_3$
Character table for $D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.972.2 x3, 6.0.2834352.2, 9.1.76556281046292101376.8 x3

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

Sibling fields

Degree 9 sibling: data not computed

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
2Data not computed
3Data not computed
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$