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Properties

Label 18.0.35194614989...9791.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{12}\cdot 37^{15}$
Root discriminant $385.40$
Ramified primes $3, 7, 37$
Class number $3293190432$ (GRH)
Class group $[6, 126, 4356072]$ (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![1509470846976, -452406256128, 545791014144, -101131530336, 65580672912, -3908734704, 3750331288, -98215254, 191614905, -21399912, 9112317, -884196, 299506, -20664, 4842, -342, 69, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 69*x^16 - 342*x^15 + 4842*x^14 - 20664*x^13 + 299506*x^12 - 884196*x^11 + 9112317*x^10 - 21399912*x^9 + 191614905*x^8 - 98215254*x^7 + 3750331288*x^6 - 3908734704*x^5 + 65580672912*x^4 - 101131530336*x^3 + 545791014144*x^2 - 452406256128*x + 1509470846976)
 
gp: K = bnfinit(x^18 + 69*x^16 - 342*x^15 + 4842*x^14 - 20664*x^13 + 299506*x^12 - 884196*x^11 + 9112317*x^10 - 21399912*x^9 + 191614905*x^8 - 98215254*x^7 + 3750331288*x^6 - 3908734704*x^5 + 65580672912*x^4 - 101131530336*x^3 + 545791014144*x^2 - 452406256128*x + 1509470846976, 1)
 

Normalized defining polynomial

\( x^{18} + 69 x^{16} - 342 x^{15} + 4842 x^{14} - 20664 x^{13} + 299506 x^{12} - 884196 x^{11} + 9112317 x^{10} - 21399912 x^{9} + 191614905 x^{8} - 98215254 x^{7} + 3750331288 x^{6} - 3908734704 x^{5} + 65580672912 x^{4} - 101131530336 x^{3} + 545791014144 x^{2} - 452406256128 x + 1509470846976 \)

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:  \(-35194614989005626327530535731700788800837489791=-\,3^{27}\cdot 7^{12}\cdot 37^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $385.40$
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$$\chi_{2331}(1,·)$, $\chi_{2331}(898,·)$, $\chi_{2331}(1220,·)$, $\chi_{2331}(1222,·)$, $\chi_{2331}(2321,·)$, $\chi_{2331}(788,·)$, $\chi_{2331}(1877,·)$, $\chi_{2331}(344,·)$, $\chi_{2331}(1444,·)$, $\chi_{2331}(988,·)$, $\chi_{2331}(100,·)$, $\chi_{2331}(2209,·)$, $\chi_{2331}(676,·)$, $\chi_{2331}(1766,·)$, $\chi_{2331}(233,·)$, $\chi_{2331}(1775,·)$, $\chi_{2331}(1331,·)$, $\chi_{2331}(1786,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{640} a^{9} + \frac{1}{320} a^{7} - \frac{1}{40} a^{6} - \frac{31}{640} a^{5} - \frac{17}{160} a^{3} + \frac{1}{40} a^{2} + \frac{7}{20} a - \frac{2}{5}$, $\frac{1}{1280} a^{10} - \frac{1}{1280} a^{9} + \frac{1}{640} a^{8} - \frac{9}{640} a^{7} - \frac{3}{256} a^{6} - \frac{49}{1280} a^{5} + \frac{3}{320} a^{4} - \frac{79}{320} a^{3} + \frac{1}{10} a^{2} + \frac{1}{5}$, $\frac{1}{2560} a^{11} - \frac{1}{2560} a^{10} - \frac{1}{1280} a^{9} + \frac{1}{1280} a^{8} + \frac{57}{2560} a^{7} + \frac{11}{512} a^{6} + \frac{17}{320} a^{5} + \frac{3}{320} a^{4} + \frac{11}{160} a^{2} - \frac{3}{8} a + \frac{2}{5}$, $\frac{1}{10240} a^{12} - \frac{1}{10240} a^{10} + \frac{1}{5120} a^{9} - \frac{37}{10240} a^{8} + \frac{1}{2560} a^{7} + \frac{217}{10240} a^{6} + \frac{49}{5120} a^{5} + \frac{147}{2560} a^{4} + \frac{259}{1280} a^{3} - \frac{9}{40} a^{2} - \frac{1}{16} a - \frac{1}{5}$, $\frac{1}{40960} a^{13} - \frac{1}{20480} a^{12} - \frac{1}{8192} a^{11} + \frac{7}{40960} a^{9} - \frac{13}{20480} a^{8} + \frac{509}{40960} a^{7} + \frac{203}{10240} a^{6} + \frac{33}{2560} a^{5} + \frac{79}{2560} a^{4} + \frac{399}{2560} a^{3} + \frac{11}{80} a^{2} + \frac{1}{32} a$, $\frac{1}{163840} a^{14} + \frac{1}{163840} a^{13} + \frac{1}{163840} a^{12} + \frac{1}{163840} a^{11} - \frac{21}{163840} a^{10} + \frac{51}{163840} a^{9} + \frac{19}{163840} a^{8} - \frac{4253}{163840} a^{7} + \frac{179}{10240} a^{6} + \frac{847}{20480} a^{5} - \frac{19}{10240} a^{4} + \frac{591}{10240} a^{3} + \frac{29}{320} a^{2} + \frac{301}{640} a + \frac{7}{20}$, $\frac{1}{119603200} a^{15} - \frac{351}{119603200} a^{14} - \frac{55}{4784128} a^{13} + \frac{1}{119603200} a^{12} + \frac{203}{119603200} a^{11} + \frac{22003}{119603200} a^{10} - \frac{74829}{119603200} a^{9} - \frac{104733}{119603200} a^{8} - \frac{187971}{7475200} a^{7} - \frac{368861}{14950400} a^{6} + \frac{294689}{7475200} a^{5} - \frac{216921}{7475200} a^{4} + \frac{22083}{93440} a^{3} + \frac{87773}{467200} a^{2} + \frac{2371}{29200} a - \frac{497}{1825}$, $\frac{1}{32053657600} a^{16} - \frac{13}{3205365760} a^{15} + \frac{7091}{8013414400} a^{14} - \frac{63741}{8013414400} a^{13} - \frac{279013}{16026828800} a^{12} - \frac{3287}{62604800} a^{11} + \frac{2487781}{8013414400} a^{10} + \frac{4469757}{8013414400} a^{9} - \frac{49705519}{32053657600} a^{8} + \frac{388346153}{16026828800} a^{7} + \frac{27639617}{4006707200} a^{6} + \frac{80776993}{2003353600} a^{5} + \frac{32494089}{2003353600} a^{4} + \frac{96626049}{1001676800} a^{3} + \frac{18964189}{125209600} a^{2} + \frac{14178587}{62604800} a + \frac{489577}{1956400}$, $\frac{1}{252525293536136008038947574007048583507136112230400} a^{17} - \frac{384089066552190511709562896516569991591}{50505058707227201607789514801409716701427222446080} a^{16} - \frac{93066769415476883193899448155985025023129}{126262646768068004019473787003524291753568056115200} a^{15} - \frac{225267782185886154289664941299313041526133}{3945707711502125125608555843860134117299001753600} a^{14} + \frac{470813300882440147159425633133626238819493717}{126262646768068004019473787003524291753568056115200} a^{13} + \frac{4463505963877529440878039203034358222617151077}{126262646768068004019473787003524291753568056115200} a^{12} - \frac{11287537586177147802517994681338936735337519163}{63131323384034002009736893501762145876784028057600} a^{11} + \frac{233944877453026605053895656709949620655964831}{7891415423004250251217111687720268234598003507200} a^{10} + \frac{57851465148910332586066352731677949286443758429}{252525293536136008038947574007048583507136112230400} a^{9} - \frac{628821367200786837728837062684603296700006538943}{252525293536136008038947574007048583507136112230400} a^{8} - \frac{1545835820846900022112764974019614288926640079781}{126262646768068004019473787003524291753568056115200} a^{7} + \frac{159555758394375583843983627005921323870116484193}{31565661692017001004868446750881072938392014028800} a^{6} + \frac{91341186072078105365603740609697854226074567137}{1972853855751062562804277921930067058649500876800} a^{5} + \frac{127846598497523103538748574003628059110966894693}{3156566169201700100486844675088107293839201402880} a^{4} - \frac{13587117757781748448314378829991806490742530961}{108101581137044523989275502571510523761616486400} a^{3} - \frac{2673421090005184518603725954406686099198516781}{14722789968291511662718491954701992974996275200} a^{2} - \frac{228136028392560760746356319276526620179742577147}{493213463937765640701069480482516764662375219200} a + \frac{1345891606065775673664065223510458655517137591}{15412920748055176271908421265078648895699225600}$

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

Class group and class number

$C_{6}\times C_{126}\times C_{4356072}$, which has order $3293190432$ (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

\(\Q(\sqrt{-111}) \), 3.3.5433561.1, 3.3.67081.1, 3.3.110889.2, 3.3.3969.1, 6.0.3277117950620031.2, 6.0.4495360700439.3, 6.0.1364897105631.1, 6.0.2393804200599.7, 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$3.6.9.9$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.9$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.9$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$[2]_{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
37Data not computed